8simplex 
Rectified 8simplex 
Truncated 8simplex  
Cantellated 8simplex 
Runcinated 8simplex 
Stericated 8simplex  
Pentellated 8simplex 
Hexicated 8simplex 
Heptellated 8simplex  
8orthoplex 
Rectified 8orthoplex 
Truncated 8orthoplex  
Cantellated 8orthoplex 
Runcinated 8orthoplex  
Hexicated 8orthoplex 
Cantellated 8cube  
Runcinated 8cube 
Stericated 8cube 
Pentellated 8cube  
Hexicated 8cube 
Heptellated 8cube  
8cube 
Rectified 8cube 
Truncated 8cube  
8demicube 
Truncated 8demicube 
Cantellated 8demicube  
Runcinated 8demicube 
Stericated 8demicube  
Pentellated 8demicube 
Hexicated 8demicube  
4_{21} 
1_{42} 
2_{41} 
In eightdimensional geometry, an eightdimensional polytope or 8polytope is a polytope contained by 7polytope facets. Each 6polytope ridge being shared by exactly two 7polytope facets.
A uniform 8polytope is one which is vertextransitive, and constructed from uniform 7polytope facets.
Regular 8polytopes can be represented by the Schläfli symbol {p,q,r,s,t,u,v}, with v {p,q,r,s,t,u} 7polytope facets around each peak.
There are exactly three such convex regular 8polytopes:
There are no nonconvex regular 8polytopes.
The topology of any given 8polytope is defined by its Betti numbers and torsion coefficients.^{[1]}
The value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions, and is zero for all 8polytopes, whatever their underlying topology. This inadequacy of the Euler characteristic to reliably distinguish between different topologies in higher dimensions led to the discovery of the more sophisticated Betti numbers.^{[1]}
Similarly, the notion of orientability of a polyhedron is insufficient to characterise the surface twistings of toroidal polytopes, and this led to the use of torsion coefficients.^{[1]}
Uniform 8polytopes with reflective symmetry can be generated by these four Coxeter groups, represented by permutations of rings of the CoxeterDynkin diagrams:
#  Coxeter group  Forms  

1  A_{8}  [3^{7}]  135  
2  BC_{8}  [4,3^{6}]  255  
3  D_{8}  [3^{5,1,1}]  191 (64 unique)  
4  E_{8}  [3^{4,2,1}]  255 
Selected regular and uniform 8polytopes from each family include:
There are many uniform prismatic families, including:
Uniform 8polytope prism families  

#  Coxeter group  CoxeterDynkin diagram  
7+1  
1  A_{7}A_{1}  [3,3,3,3,3,3]×[ ]  
2  B_{7}A_{1}  [4,3,3,3,3,3]×[ ]  
3  D_{7}A_{1}  [3^{4,1,1}]×[ ]  
4  E_{7}A_{1}  [3^{3,2,1}]×[ ]  
6+2  
1  A_{6}I_{2}(p)  [3,3,3,3,3]×[p]  
2  B_{6}I_{2}(p)  [4,3,3,3,3]×[p]  
3  D_{6}I_{2}(p)  [3^{3,1,1}]×[p]  
4  E_{6}I_{2}(p)  [3,3,3,3,3]×[p]  
6+1+1  
1  A_{6}A_{1}A_{1}  [3,3,3,3,3]×[ ]x[ ]  
2  B_{6}A_{1}A_{1}  [4,3,3,3,3]×[ ]x[ ]  
3  D_{6}A_{1}A_{1}  [3^{3,1,1}]×[ ]x[ ]  
4  E_{6}A_{1}A_{1}  [3,3,3,3,3]×[ ]x[ ]  
5+3  
1  A_{5}A_{3}  [3^{4}]×[3,3]  
2  B_{5}A_{3}  [4,3^{3}]×[3,3]  
3  D_{5}A_{3}  [3^{2,1,1}]×[3,3]  
4  A_{5}B_{3}  [3^{4}]×[4,3]  
5  B_{5}B_{3}  [4,3^{3}]×[4,3]  
6  D_{5}B_{3}  [3^{2,1,1}]×[4,3]  
7  A_{5}H_{3}  [3^{4}]×[5,3]  
8  B_{5}H_{3}  [4,3^{3}]×[5,3]  
9  D_{5}H_{3}  [3^{2,1,1}]×[5,3]  
5+2+1  
1  A_{5}I_{2}(p)A_{1}  [3,3,3]×[p]×[ ]  
2  B_{5}I_{2}(p)A_{1}  [4,3,3]×[p]×[ ]  
3  D_{5}I_{2}(p)A_{1}  [3^{2,1,1}]×[p]×[ ]  
5+1+1+1  
1  A_{5}A_{1}A_{1}A_{1}  [3,3,3]×[ ]×[ ]×[ ]  
2  B_{5}A_{1}A_{1}A_{1}  [4,3,3]×[ ]×[ ]×[ ]  
3  D_{5}A_{1}A_{1}A_{1}  [3^{2,1,1}]×[ ]×[ ]×[ ]  
4+4  
1  A_{4}A_{4}  [3,3,3]×[3,3,3]  
2  B_{4}A_{4}  [4,3,3]×[3,3,3]  
3  D_{4}A_{4}  [3^{1,1,1}]×[3,3,3]  
4  F_{4}A_{4}  [3,4,3]×[3,3,3]  
5  H_{4}A_{4}  [5,3,3]×[3,3,3]  
6  B_{4}B_{4}  [4,3,3]×[4,3,3]  
7  D_{4}B_{4}  [3^{1,1,1}]×[4,3,3]  
8  F_{4}B_{4}  [3,4,3]×[4,3,3]  
9  H_{4}B_{4}  [5,3,3]×[4,3,3]  
10  D_{4}D_{4}  [3^{1,1,1}]×[3^{1,1,1}]  
11  F_{4}D_{4}  [3,4,3]×[3^{1,1,1}]  
12  H_{4}D_{4}  [5,3,3]×[3^{1,1,1}]  
13  F_{4}×F_{4}  [3,4,3]×[3,4,3]  
14  H_{4}×F_{4}  [5,3,3]×[3,4,3]  
15  H_{4}H_{4}  [5,3,3]×[5,3,3]  
4+3+1  
1  A_{4}A_{3}A_{1}  [3,3,3]×[3,3]×[ ]  
2  A_{4}B_{3}A_{1}  [3,3,3]×[4,3]×[ ]  
3  A_{4}H_{3}A_{1}  [3,3,3]×[5,3]×[ ]  
4  B_{4}A_{3}A_{1}  [4,3,3]×[3,3]×[ ]  
5  B_{4}B_{3}A_{1}  [4,3,3]×[4,3]×[ ]  
6  B_{4}H_{3}A_{1}  [4,3,3]×[5,3]×[ ]  
7  H_{4}A_{3}A_{1}  [5,3,3]×[3,3]×[ ]  
8  H_{4}B_{3}A_{1}  [5,3,3]×[4,3]×[ ]  
9  H_{4}H_{3}A_{1}  [5,3,3]×[5,3]×[ ]  
10  F_{4}A_{3}A_{1}  [3,4,3]×[3,3]×[ ]  
11  F_{4}B_{3}A_{1}  [3,4,3]×[4,3]×[ ]  
12  F_{4}H_{3}A_{1}  [3,4,3]×[5,3]×[ ]  
13  D_{4}A_{3}A_{1}  [3^{1,1,1}]×[3,3]×[ ]  
14  D_{4}B_{3}A_{1}  [3^{1,1,1}]×[4,3]×[ ]  
15  D_{4}H_{3}A_{1}  [3^{1,1,1}]×[5,3]×[ ]  
4+2+2  
...  
4+2+1+1  
...  
4+1+1+1+1  
...  
3+3+2  
1  A_{3}A_{3}I_{2}(p)  [3,3]×[3,3]×[p]  
2  B_{3}A_{3}I_{2}(p)  [4,3]×[3,3]×[p]  
3  H_{3}A_{3}I_{2}(p)  [5,3]×[3,3]×[p]  
4  B_{3}B_{3}I_{2}(p)  [4,3]×[4,3]×[p]  
5  H_{3}B_{3}I_{2}(p)  [5,3]×[4,3]×[p]  
6  H_{3}H_{3}I_{2}(p)  [5,3]×[5,3]×[p]  
3+3+1+1  
1  A_{3}^{2}A_{1}^{2}  [3,3]×[3,3]×[ ]×[ ]  
2  B_{3}A_{3}A_{1}^{2}  [4,3]×[3,3]×[ ]×[ ]  
3  H_{3}A_{3}A_{1}^{2}  [5,3]×[3,3]×[ ]×[ ]  
4  B_{3}B_{3}A_{1}^{2}  [4,3]×[4,3]×[ ]×[ ]  
5  H_{3}B_{3}A_{1}^{2}  [5,3]×[4,3]×[ ]×[ ]  
6  H_{3}H_{3}A_{1}^{2}  [5,3]×[5,3]×[ ]×[ ]  
3+2+2+1  
1  A_{3}I_{2}(p)I_{2}(q)A_{1}  [3,3]×[p]×[q]×[ ]  
2  B_{3}I_{2}(p)I_{2}(q)A_{1}  [4,3]×[p]×[q]×[ ]  
3  H_{3}I_{2}(p)I_{2}(q)A_{1}  [5,3]×[p]×[q]×[ ]  
3+2+1+1+1  
1  A_{3}I_{2}(p)A_{1}^{3}  [3,3]×[p]×[ ]x[ ]×[ ]  
2  B_{3}I_{2}(p)A_{1}^{3}  [4,3]×[p]×[ ]x[ ]×[ ]  
3  H_{3}I_{2}(p)A_{1}^{3}  [5,3]×[p]×[ ]x[ ]×[ ]  
3+1+1+1+1+1  
1  A_{3}A_{1}^{5}  [3,3]×[ ]x[ ]×[ ]x[ ]×[ ]  
2  B_{3}A_{1}^{5}  [4,3]×[ ]x[ ]×[ ]x[ ]×[ ]  
3  H_{3}A_{1}^{5}  [5,3]×[ ]x[ ]×[ ]x[ ]×[ ]  
2+2+2+2  
1  I_{2}(p)I_{2}(q)I_{2}(r)I_{2}(s)  [p]×[q]×[r]×[s]  
2+2+2+1+1  
1  I_{2}(p)I_{2}(q)I_{2}(r)A_{1}^{2}  [p]×[q]×[r]×[ ]×[ ]  
2+2+1+1+1+1  
2  I_{2}(p)I_{2}(q)A_{1}^{4}  [p]×[q]×[ ]×[ ]×[ ]×[ ]  
2+1+1+1+1+1+1  
1  I_{2}(p)A_{1}^{6}  [p]×[ ]×[ ]×[ ]×[ ]×[ ]×[ ]  
1+1+1+1+1+1+1+1  
1  A_{1}^{8}  [ ]×[ ]×[ ]×[ ]×[ ]×[ ]×[ ]×[ ] 
The A_{8} family has symmetry of order 362880 (9 factorial).
There are 135 forms based on all permutations of the CoxeterDynkin diagrams with one or more rings. (128+81 cases) These are all enumerated below. Bowersstyle acronym names are given in parentheses for crossreferencing.
See also a list of 8simplex polytopes for symmetric Coxeter plane graphs of these polytopes.
A_{8} uniform polytopes  

#  CoxeterDynkin diagram  Truncation indices 
Johnson name  Basepoint  Element counts  
7  6  5  4  3  2  1  0  
1 

t_{0}  8simplex (ene)  (0,0,0,0,0,0,0,0,1)  9  36  84  126  126  84  36  9 
2 

t_{1}  Rectified 8simplex (rene)  (0,0,0,0,0,0,0,1,1)  18  108  336  630  576  588  252  36 
3 

t_{2}  Birectified 8simplex (bene)  (0,0,0,0,0,0,1,1,1)  18  144  588  1386  2016  1764  756  84 
4 

t_{3}  Trirectified 8simplex (trene)  (0,0,0,0,0,1,1,1,1)  1260  126  
5 

t_{0,1}  Truncated 8simplex (tene)  (0,0,0,0,0,0,0,1,2)  288  72  
6 

t_{0,2}  Cantellated 8simplex  (0,0,0,0,0,0,1,1,2)  1764  252  
7 

t_{1,2}  Bitruncated 8simplex  (0,0,0,0,0,0,1,2,2)  1008  252  
8 

t_{0,3}  Runcinated 8simplex  (0,0,0,0,0,1,1,1,2)  4536  504  
9 

t_{1,3}  Bicantellated 8simplex  (0,0,0,0,0,1,1,2,2)  5292  756  
10 

t_{2,3}  Tritruncated 8simplex  (0,0,0,0,0,1,2,2,2)  2016  504  
11 

t_{0,4}  Stericated 8simplex  (0,0,0,0,1,1,1,1,2)  6300  630  
12 

t_{1,4}  Biruncinated 8simplex  (0,0,0,0,1,1,1,2,2)  11340  1260  
13 

t_{2,4}  Tricantellated 8simplex  (0,0,0,0,1,1,2,2,2)  8820  1260  
14 

t_{3,4}  Quadritruncated 8simplex  (0,0,0,0,1,2,2,2,2)  2520  630  
15 

t_{0,5}  Pentellated 8simplex  (0,0,0,1,1,1,1,1,2)  5040  504  
16 

t_{1,5}  Bistericated 8simplex  (0,0,0,1,1,1,1,2,2)  12600  1260  
17 

t_{2,5}  Triruncinated 8simplex  (0,0,0,1,1,1,2,2,2)  15120  1680  
18 

t_{0,6}  Hexicated 8simplex  (0,0,1,1,1,1,1,1,2)  2268  252  
19 

t_{1,6}  Bipentellated 8simplex  (0,0,1,1,1,1,1,2,2)  7560  756  
20 

t_{0,7}  Heptellated 8simplex  (0,1,1,1,1,1,1,1,2)  504  72  
21 

t_{0,1,2}  Cantitruncated 8simplex  (0,0,0,0,0,0,1,2,3)  2016  504  
22 

t_{0,1,3}  Runcitruncated 8simplex  (0,0,0,0,0,1,1,2,3)  9828  1512  
23 

t_{0,2,3}  Runcicantellated 8simplex  (0,0,0,0,0,1,2,2,3)  6804  1512  
24 

t_{1,2,3}  Bicantitruncated 8simplex  (0,0,0,0,0,1,2,3,3)  6048  1512  
25 

t_{0,1,4}  Steritruncated 8simplex  (0,0,0,0,1,1,1,2,3)  20160  2520  
26 

t_{0,2,4}  Stericantellated 8simplex  (0,0,0,0,1,1,2,2,3)  26460  3780  
27 

t_{1,2,4}  Biruncitruncated 8simplex  (0,0,0,0,1,1,2,3,3)  22680  3780  
28 

t_{0,3,4}  Steriruncinated 8simplex  (0,0,0,0,1,2,2,2,3)  12600  2520  
29 

t_{1,3,4}  Biruncicantellated 8simplex  (0,0,0,0,1,2,2,3,3)  18900  3780  
30 

t_{2,3,4}  Tricantitruncated 8simplex  (0,0,0,0,1,2,3,3,3)  10080  2520  
31 

t_{0,1,5}  Pentitruncated 8simplex  (0,0,0,1,1,1,1,2,3)  21420  2520  
32 

t_{0,2,5}  Penticantellated 8simplex  (0,0,0,1,1,1,2,2,3)  42840  5040  
33 

t_{1,2,5}  Bisteritruncated 8simplex  (0,0,0,1,1,1,2,3,3)  35280  5040  
34 

t_{0,3,5}  Pentiruncinated 8simplex  (0,0,0,1,1,2,2,2,3)  37800  5040  
35 

t_{1,3,5}  Bistericantellated 8simplex  (0,0,0,1,1,2,2,3,3)  52920  7560  
36 

t_{2,3,5}  Triruncitruncated 8simplex  (0,0,0,1,1,2,3,3,3)  27720  5040  
37 

t_{0,4,5}  Pentistericated 8simplex  (0,0,0,1,2,2,2,2,3)  13860  2520  
38 

t_{1,4,5}  Bisteriruncinated 8simplex  (0,0,0,1,2,2,2,3,3)  30240  5040  
39 

t_{0,1,6}  Hexitruncated 8simplex  (0,0,1,1,1,1,1,2,3)  12096  1512  
40 

t_{0,2,6}  Hexicantellated 8simplex  (0,0,1,1,1,1,2,2,3)  34020  3780  
41 

t_{1,2,6}  Bipentitruncated 8simplex  (0,0,1,1,1,1,2,3,3)  26460  3780  
42 

t_{0,3,6}  Hexiruncinated 8simplex  (0,0,1,1,1,2,2,2,3)  45360  5040  
43 

t_{1,3,6}  Bipenticantellated 8simplex  (0,0,1,1,1,2,2,3,3)  60480  7560  
44 

t_{0,4,6}  Hexistericated 8simplex  (0,0,1,1,2,2,2,2,3)  30240  3780  
45 

t_{0,5,6}  Hexipentellated 8simplex  (0,0,1,2,2,2,2,2,3)  9072  1512  
46 

t_{0,1,7}  Heptitruncated 8simplex  (0,1,1,1,1,1,1,2,3)  3276  504  
47 

t_{0,2,7}  Hepticantellated 8simplex  (0,1,1,1,1,1,2,2,3)  12852  1512  
48 

t_{0,3,7}  Heptiruncinated 8simplex  (0,1,1,1,1,2,2,2,3)  23940  2520  
49 

t_{0,1,2,3}  Runcicantitruncated 8simplex  (0,0,0,0,0,1,2,3,4)  12096  3024  
50 

t_{0,1,2,4}  Stericantitruncated 8simplex  (0,0,0,0,1,1,2,3,4)  45360  7560  
51 

t_{0,1,3,4}  Steriruncitruncated 8simplex  (0,0,0,0,1,2,2,3,4)  34020  7560  
52 

t_{0,2,3,4}  Steriruncicantellated 8simplex  (0,0,0,0,1,2,3,3,4)  34020  7560  
53 

t_{1,2,3,4}  Biruncicantitruncated 8simplex  (0,0,0,0,1,2,3,4,4)  30240  7560  
54 

t_{0,1,2,5}  Penticantitruncated 8simplex  (0,0,0,1,1,1,2,3,4)  70560  10080  
55 

t_{0,1,3,5}  Pentiruncitruncated 8simplex  (0,0,0,1,1,2,2,3,4)  98280  15120  
56 

t_{0,2,3,5}  Pentiruncicantellated 8simplex  (0,0,0,1,1,2,3,3,4)  90720  15120  
57 

t_{1,2,3,5}  Bistericantitruncated 8simplex  (0,0,0,1,1,2,3,4,4)  83160  15120  
58 

t_{0,1,4,5}  Pentisteritruncated 8simplex  (0,0,0,1,2,2,2,3,4)  50400  10080  
59 

t_{0,2,4,5}  Pentistericantellated 8simplex  (0,0,0,1,2,2,3,3,4)  83160  15120  
60 

t_{1,2,4,5}  Bisteriruncitruncated 8simplex  (0,0,0,1,2,2,3,4,4)  68040  15120  
61 

t_{0,3,4,5}  Pentisteriruncinated 8simplex  (0,0,0,1,2,3,3,3,4)  50400  10080  
62 

t_{1,3,4,5}  Bisteriruncicantellated 8simplex  (0,0,0,1,2,3,3,4,4)  75600  15120  
63 

t_{2,3,4,5}  Triruncicantitruncated 8simplex  (0,0,0,1,2,3,4,4,4)  40320  10080  
64 

t_{0,1,2,6}  Hexicantitruncated 8simplex  (0,0,1,1,1,1,2,3,4)  52920  7560  
65 

t_{0,1,3,6}  Hexiruncitruncated 8simplex  (0,0,1,1,1,2,2,3,4)  113400  15120  
66 

t_{0,2,3,6}  Hexiruncicantellated 8simplex  (0,0,1,1,1,2,3,3,4)  98280  15120  
67 

t_{1,2,3,6}  Bipenticantitruncated 8simplex  (0,0,1,1,1,2,3,4,4)  90720  15120  
68 

t_{0,1,4,6}  Hexisteritruncated 8simplex  (0,0,1,1,2,2,2,3,4)  105840  15120  
69 

t_{0,2,4,6}  Hexistericantellated 8simplex  (0,0,1,1,2,2,3,3,4)  158760  22680  
70 

t_{1,2,4,6}  Bipentiruncitruncated 8simplex  (0,0,1,1,2,2,3,4,4)  136080  22680  
71 

t_{0,3,4,6}  Hexisteriruncinated 8simplex  (0,0,1,1,2,3,3,3,4)  90720  15120  
72 

t_{1,3,4,6}  Bipentiruncicantellated 8simplex  (0,0,1,1,2,3,3,4,4)  136080  22680  
73 

t_{0,1,5,6}  Hexipentitruncated 8simplex  (0,0,1,2,2,2,2,3,4)  41580  7560  
74 

t_{0,2,5,6}  Hexipenticantellated 8simplex  (0,0,1,2,2,2,3,3,4)  98280  15120  
75 

t_{1,2,5,6}  Bipentisteritruncated 8simplex  (0,0,1,2,2,2,3,4,4)  75600  15120  
76 

t_{0,3,5,6}  Hexipentiruncinated 8simplex  (0,0,1,2,2,3,3,3,4)  98280  15120  
77 

t_{0,4,5,6}  Hexipentistericated 8simplex  (0,0,1,2,3,3,3,3,4)  41580  7560  
78 

t_{0,1,2,7}  Hepticantitruncated 8simplex  (0,1,1,1,1,1,2,3,4)  18144  3024  
79 

t_{0,1,3,7}  Heptiruncitruncated 8simplex  (0,1,1,1,1,2,2,3,4)  56700  7560  
80 

t_{0,2,3,7}  Heptiruncicantellated 8simplex  (0,1,1,1,1,2,3,3,4)  45360  7560  
81 

t_{0,1,4,7}  Heptisteritruncated 8simplex  (0,1,1,1,2,2,2,3,4)  80640  10080  
82 

t_{0,2,4,7}  Heptistericantellated 8simplex  (0,1,1,1,2,2,3,3,4)  113400  15120  
83 

t_{0,3,4,7}  Heptisteriruncinated 8simplex  (0,1,1,1,2,3,3,3,4)  60480  10080  
84 

t_{0,1,5,7}  Heptipentitruncated 8simplex  (0,1,1,2,2,2,2,3,4)  56700  7560  
85 

t_{0,2,5,7}  Heptipenticantellated 8simplex  (0,1,1,2,2,2,3,3,4)  120960  15120  
86 

t_{0,1,6,7}  Heptihexitruncated 8simplex  (0,1,2,2,2,2,2,3,4)  18144  3024  
87 

t_{0,1,2,3,4}  Steriruncicantitruncated 8simplex  (0,0,0,0,1,2,3,4,5)  60480  15120  
88 

t_{0,1,2,3,5}  Pentiruncicantitruncated 8simplex  (0,0,0,1,1,2,3,4,5)  166320  30240  
89 

t_{0,1,2,4,5}  Pentistericantitruncated 8simplex  (0,0,0,1,2,2,3,4,5)  136080  30240  
90 

t_{0,1,3,4,5}  Pentisteriruncitruncated 8simplex  (0,0,0,1,2,3,3,4,5)  136080  30240  
91 

t_{0,2,3,4,5}  Pentisteriruncicantellated 8simplex  (0,0,0,1,2,3,4,4,5)  136080  30240  
92 

t_{1,2,3,4,5}  Bisteriruncicantitruncated 8simplex  (0,0,0,1,2,3,4,5,5)  120960  30240  
93 

t_{0,1,2,3,6}  Hexiruncicantitruncated 8simplex  (0,0,1,1,1,2,3,4,5)  181440  30240  
94 

t_{0,1,2,4,6}  Hexistericantitruncated 8simplex  (0,0,1,1,2,2,3,4,5)  272160  45360  
95 

t_{0,1,3,4,6}  Hexisteriruncitruncated 8simplex  (0,0,1,1,2,3,3,4,5)  249480  45360  
96 

t_{0,2,3,4,6}  Hexisteriruncicantellated 8simplex  (0,0,1,1,2,3,4,4,5)  249480  45360  
97 

t_{1,2,3,4,6}  Bipentiruncicantitruncated 8simplex  (0,0,1,1,2,3,4,5,5)  226800  45360  
98 

t_{0,1,2,5,6}  Hexipenticantitruncated 8simplex  (0,0,1,2,2,2,3,4,5)  151200  30240  
99 

t_{0,1,3,5,6}  Hexipentiruncitruncated 8simplex  (0,0,1,2,2,3,3,4,5)  249480  45360  
100 

t_{0,2,3,5,6}  Hexipentiruncicantellated 8simplex  (0,0,1,2,2,3,4,4,5)  226800  45360  
101 

t_{1,2,3,5,6}  Bipentistericantitruncated 8simplex  (0,0,1,2,2,3,4,5,5)  204120  45360  
102 

t_{0,1,4,5,6}  Hexipentisteritruncated 8simplex  (0,0,1,2,3,3,3,4,5)  151200  30240  
103 

t_{0,2,4,5,6}  Hexipentistericantellated 8simplex  (0,0,1,2,3,3,4,4,5)  249480  45360  
104 

t_{0,3,4,5,6}  Hexipentisteriruncinated 8simplex  (0,0,1,2,3,4,4,4,5)  151200  30240  
105 

t_{0,1,2,3,7}  Heptiruncicantitruncated 8simplex  (0,1,1,1,1,2,3,4,5)  83160  15120  
106 

t_{0,1,2,4,7}  Heptistericantitruncated 8simplex  (0,1,1,1,2,2,3,4,5)  196560  30240  
107 

t_{0,1,3,4,7}  Heptisteriruncitruncated 8simplex  (0,1,1,1,2,3,3,4,5)  166320  30240  
108 

t_{0,2,3,4,7}  Heptisteriruncicantellated 8simplex  (0,1,1,1,2,3,4,4,5)  166320  30240  
109 

t_{0,1,2,5,7}  Heptipenticantitruncated 8simplex  (0,1,1,2,2,2,3,4,5)  196560  30240  
110 

t_{0,1,3,5,7}  Heptipentiruncitruncated 8simplex  (0,1,1,2,2,3,3,4,5)  294840  45360  
111 

t_{0,2,3,5,7}  Heptipentiruncicantellated 8simplex  (0,1,1,2,2,3,4,4,5)  272160  45360  
112 

t_{0,1,4,5,7}  Heptipentisteritruncated 8simplex  (0,1,1,2,3,3,3,4,5)  166320  30240  
113 

t_{0,1,2,6,7}  Heptihexicantitruncated 8simplex  (0,1,2,2,2,2,3,4,5)  83160  15120  
114 

t_{0,1,3,6,7}  Heptihexiruncitruncated 8simplex  (0,1,2,2,2,3,3,4,5)  196560  30240  
115 

t_{0,1,2,3,4,5}  Pentisteriruncicantitruncated 8simplex  (0,0,0,1,2,3,4,5,6)  241920  60480  
116 

t_{0,1,2,3,4,6}  Hexisteriruncicantitruncated 8simplex  (0,0,1,1,2,3,4,5,6)  453600  90720  
117 

t_{0,1,2,3,5,6}  Hexipentiruncicantitruncated 8simplex  (0,0,1,2,2,3,4,5,6)  408240  90720  
118 

t_{0,1,2,4,5,6}  Hexipentistericantitruncated 8simplex  (0,0,1,2,3,3,4,5,6)  408240  90720  
119 

t_{0,1,3,4,5,6}  Hexipentisteriruncitruncated 8simplex  (0,0,1,2,3,4,4,5,6)  408240  90720  
120 

t_{0,2,3,4,5,6}  Hexipentisteriruncicantellated 8simplex  (0,0,1,2,3,4,5,5,6)  408240  90720  
121 

t_{1,2,3,4,5,6}  Bipentisteriruncicantitruncated 8simplex  (0,0,1,2,3,4,5,6,6)  362880  90720  
122 

t_{0,1,2,3,4,7}  Heptisteriruncicantitruncated 8simplex  (0,1,1,1,2,3,4,5,6)  302400  60480  
123 

t_{0,1,2,3,5,7}  Heptipentiruncicantitruncated 8simplex  (0,1,1,2,2,3,4,5,6)  498960  90720  
124 

t_{0,1,2,4,5,7}  Heptipentistericantitruncated 8simplex  (0,1,1,2,3,3,4,5,6)  453600  90720  
125 

t_{0,1,3,4,5,7}  Heptipentisteriruncitruncated 8simplex  (0,1,1,2,3,4,4,5,6)  453600  90720  
126 

t_{0,2,3,4,5,7}  Heptipentisteriruncicantellated 8simplex  (0,1,1,2,3,4,5,5,6)  453600  90720  
127 

t_{0,1,2,3,6,7}  Heptihexiruncicantitruncated 8simplex  (0,1,2,2,2,3,4,5,6)  302400  60480  
128 

t_{0,1,2,4,6,7}  Heptihexistericantitruncated 8simplex  (0,1,2,2,3,3,4,5,6)  498960  90720  
129 

t_{0,1,3,4,6,7}  Heptihexisteriruncitruncated 8simplex  (0,1,2,2,3,4,4,5,6)  453600  90720  
130 

t_{0,1,2,5,6,7}  Heptihexipenticantitruncated 8simplex  (0,1,2,3,3,3,4,5,6)  302400  60480  
131 

t_{0,1,2,3,4,5,6}  Hexipentisteriruncicantitruncated 8simplex  (0,0,1,2,3,4,5,6,7)  725760  181440  
132 

t_{0,1,2,3,4,5,7}  Heptipentisteriruncicantitruncated 8simplex  (0,1,1,2,3,4,5,6,7)  816480  181440  
133 

t_{0,1,2,3,4,6,7}  Heptihexisteriruncicantitruncated 8simplex  (0,1,2,2,3,4,5,6,7)  816480  181440  
134 

t_{0,1,2,3,5,6,7}  Heptihexipentiruncicantitruncated 8simplex  (0,1,2,3,3,4,5,6,7)  816480  181440  
135 

t_{0,1,2,3,4,5,6,7}  Omnitruncated 8simplex  (0,1,2,3,4,5,6,7,8)  1451520  362880 
The B_{8} family has symmetry of order 10321920 (8 factorial x 2^{8}). There are 255 forms based on all permutations of the CoxeterDynkin diagrams with one or more rings.
See also a list of B8 polytopes for symmetric Coxeter plane graphs of these polytopes.
B_{8} uniform polytopes  

#  CoxeterDynkin diagram  Schläfli symbol 
Name  Element counts  
7  6  5  4  3  2  1  0  
1  t_{0}{3^{6},4}  8orthoplex Diacosipentacontahexazetton (ek) 
256  1024  1792  1792  1120  448  112  16  
2  t_{1}{3^{6},4}  Rectified 8orthoplex Rectified diacosipentacontahexazetton (rek) 
272  3072  8960  12544  10080  4928  1344  112  
3  t_{2}{3^{6},4}  Birectified 8orthoplex Birectified diacosipentacontahexazetton (bark) 
272  3184  16128  34048  36960  22400  6720  448  
4  t_{3}{3^{6},4}  Trirectified 8orthoplex Trirectified diacosipentacontahexazetton (tark) 
272  3184  16576  48384  71680  53760  17920  1120  
5  t_{3}{4,3^{6}}  Trirectified 8cube Trirectified octeract (tro) 
272  3184  16576  47712  80640  71680  26880  1792  
6  t_{2}{4,3^{6}}  Birectified 8cube Birectified octeract (bro) 
272  3184  14784  36960  55552  50176  21504  1792  
7  t_{1}{4,3^{6}}  Rectified 8cube Rectified octeract (recto) 
272  2160  7616  15456  19712  16128  7168  1024  
8  t_{0}{4,3^{6}}  8cube Octeract (octo) 
16  112  448  1120  1792  1792  1024  256  
9  t_{0,1}{3^{6},4}  Truncated 8orthoplex Truncated diacosipentacontahexazetton (tek) 
1456  224  
10  t_{0,2}{3^{6},4}  Cantellated 8orthoplex Small rhombated diacosipentacontahexazetton (srek) 
14784  1344  
11  t_{1,2}{3^{6},4}  Bitruncated 8orthoplex Bitruncated diacosipentacontahexazetton (batek) 
8064  1344  
12  t_{0,3}{3^{6},4}  Runcinated 8orthoplex Small prismated diacosipentacontahexazetton (spek) 
60480  4480  
13  t_{1,3}{3^{6},4}  Bicantellated 8orthoplex Small birhombated diacosipentacontahexazetton (sabork) 
67200  6720  
14  t_{2,3}{3^{6},4}  Tritruncated 8orthoplex Tritruncated diacosipentacontahexazetton (tatek) 
24640  4480  
15  t_{0,4}{3^{6},4}  Stericated 8orthoplex Small cellated diacosipentacontahexazetton (scak) 
125440  8960  
16  t_{1,4}{3^{6},4}  Biruncinated 8orthoplex Small biprismated diacosipentacontahexazetton (sabpek) 
215040  17920  
17  t_{2,4}{3^{6},4}  Tricantellated 8orthoplex Small trirhombated diacosipentacontahexazetton (satrek) 
161280  17920  
18  t_{3,4}{4,3^{6}}  Quadritruncated 8cube Octeractidiacosipentacontahexazetton (oke) 
< 