Gyroelongated square bipyramid

The gyroelongated square bipyramid is one of the 92 Johnson solids (J17). It consists of 8+8 triangles as faces. It can be constructed by inserting a square antiprism between the two pyramidal halves of the regular octahedron, seen as a square bipyramid.

Vertex coordinates
A gyroelongated square bipyramid of edge length 1 has the following vertices:
 * $$\left(0,\,0,\,±\frac{2\sqrt2+\sqrt[4]{8}}{4}\right),$$
 * $$\left(±\frac12,\,±\frac12,\,\frac{\sqrt[4]{8}}{4}\right),$$
 * $$\left(0,\,±\frac{\sqrt2}{2},\,-\frac{\sqrt[4]{8}}{4}\right),$$
 * $$\left(±\frac{\sqrt2}{2},\,0,\,-\frac{\sqrt[4]{8}}{4}\right).$$