Demiocteract (8demicube)  

Petrie polygon projection  
Type  Uniform 8polytope 
Family  demihypercube 
Coxeter symbol  1_{51} 
Schläfli symbols  {3,3^{5,1}} = h{4,3^{6}} s{2^{1,1,1,1,1,1,1}} 
Coxeter diagrams  =

7faces  144: 16 {3^{1,4,1}} 128 {3^{6}} 
6faces  112 {3^{1,3,1}} 1024 {3^{5}} 
5faces  448 {3^{1,2,1}} 3584 {3^{4}} 
4faces  1120 {3^{1,1,1}} 7168 {3,3,3} 
Cells  10752: 1792 {3^{1,0,1}} 8960 {3,3} 
Faces  7168 {3} 
Edges  1792 
Vertices  128 
Vertex figure  Rectified 7simplex 
Symmetry group  D_{8}, [3^{5,1,1}] = [1^{+},4,3^{6}] A_{1}^{8}, [2^{7}]^{+} 
Dual  ? 
Properties  convex 
In geometry, a demiocteract or 8demicube is a uniform 8polytope, constructed from the 8hypercube, octeract, with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as HM_{8} for an 8dimensional half measure polytope.
Coxeter named this polytope as 1_{51} from its Coxeter diagram, with a ring on one of the 1length branches, and Schläfli symbol or {3,3^{5,1}}.
Cartesian coordinates for the vertices of an 8demicube centered at the origin are alternate halves of the 8cube:
with an odd number of plus signs.
This polytope is the vertex figure for the uniform tessellation, 2_{51} with CoxeterDynkin diagram:
Coxeter plane  B_{8}  D_{8}  D_{7}  D_{6}  D_{5} 

Graph  
Dihedral symmetry  [16/2]  [14]  [12]  [10]  [8] 
Coxeter plane  D_{4}  D_{3}  A_{7}  A_{5}  A_{3} 
Graph  
Dihedral symmetry  [6]  [4]  [8]  [6]  [4] 