Octahedron atop cube

The octahedron atop cube, or octacube, is a CRF segmentochoron (designated K-4.15 on Richard Klitzing's list). As the name suggests, it consists of a cube and an octahedron as bases, connected by 6 square pyramids and 8+12 tetrahedra.

It is also commonly referred to as a cubic or octahedral antiprism, as the two bases are a pair of dual polyhedra.

Vertex coordinates
The vertices of an octahedron atop cube segmentochoron of edge length 1 are given by:
 * $$\left(±\frac{\sqrt2}{2},\,0,\,0,\,\frac{\sqrt{2\sqrt2-1}}{2}\right)$$ and all permutations of its first 3 coordinates,
 * $$\left(±\frac12,\,±\frac12,\,±\frac12,\,0\right).$$