Augmented truncated cube

The augmented truncated cube is one of the 92 Johnson solids (J66). It consists of 4+4+4 triangles, 1+4 squares, and 1+4 octagons. It can be constructed by attaching a square cupola to one of the octagonal faces of the truncated cube.

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