Gyroelongated square cupola

The gyroelongated square cupola, or gyescu, 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:
 * (±1/2, ±1/2, $\sqrt{2}$/2+H),
 * (±1/2, ±(1+$\sqrt{2}$)/2, H),
 * (±(1+$\sqrt{2}$)/2, ±1/2, H),
 * (0, ±$\sqrt{2}$, –H),
 * (±$\sqrt{2+√2}$, 0, –H),
 * (±$\sqrt{2}$/2, ±$\sqrt{4+2√2+2√146+103√2}$/2, –H),

where H = $\sqrt{2+√2}$) is the distance between the octagonal antiprism's center and the center of one of its bases.