Biaugmented truncated cube

The biaugmented truncated cube is one of the 92 Johnson solids (J67). It consists of 8+8 triangles, 2+8 squares, and 4 octagons. It can be constructed by attaching square cupolas to two opposite octagonal faces of the truncated cube..

Vertex coordinates
A biaugmented truncated cube of edge length 1 has vertices given by all permutations of: Plus the following additional vertices:
 * $$\left(±\frac{1+\sqrt2}{2},\,±\frac{1+\sqrt2}{2},\,±\frac12\right),$$
 * $$\left(±\frac{\sqrt2}{2},\,0,\,±\frac{1+2\sqrt2}{2}\right),$$
 * $$\left(0,\,±\frac{\sqrt2}{2},\,±\frac{1+2\sqrt2}{2}\right).$$