Gyroelongated square cupola

The gyroelongated square cupola is one of the 92 Johnson solids (J23). It consists of 4+4+4+8 triangles, 1+4 squares, and 1 octagon. It can be constructed by attaching an octagonal antiprism to the octagonal base of the square cupola.

If a second cupola is attached to the other octagonal base of the antiprism, the result is the gyroelongated square bicupola.

Vertex coordinates
A gyroelongated square cupola of edge length 1 has the following vertices: where $$H=\sqrt{\frac{-2-2\sqrt2+\sqrt{20+14\sqrt2}}{8}}$$ is the distance between the octagonal antiprism's center and the center of one of its bases.
 * $$\left(±\frac12,\,±\frac12,\,\frac{\sqrt2+2H}{2}\right),$$
 * $$\left(±\frac12,\,±\frac{1+\sqrt2}2,\,H\right),$$
 * $$\left(±\frac{1+\sqrt2}2,\,±\frac12,\,H\right),$$
 * $$\left(0,\,±\sqrt{\frac{2+\sqrt2}2},\,-H\right),$$
 * $$\left(±\sqrt{\frac{2+\sqrt2}2},\,0,\,-H\right),$$
 * $$\left(±\frac{\sqrt{2+\sqrt2}}2,\,±\frac{\sqrt{2+\sqrt2}}2,\,-H\right),$$