Elongated square cupola

The elongated square cupola is one of the 92 Johnson solids (J19). It consists of 4 triangles, 1+4+4+4 squares, and 1 octagon. It can be constructed by attaching an octagonal prism to the octagonal base of the square cupola.

It can also be seen as a diminished small rhombicuboctahedron, formed by cutting off one of its square cupola segments. Conversely, attaching a second square cupola to the other octagonal base of the prism in the same orientation leads to the small rhombicuboctahedron, seen as an elongated square orthobicupola. If the second cupola is rotated by 45º instead, the result is an elongated square gyrobicupola.

Vertex coordinates
An elongated square cupola of edge length 1 has the following vertices:
 * $$\left(±\frac12,\,±\frac{1+\sqrt2}{2},\,±\frac12\right),$$
 * $$\left(±\frac{1+\sqrt2}{2},\,±\frac12,\,±\frac12\right),$$
 * $$\left(±\frac12,\,±\frac12\,\frac{1+\sqrt2}{2}\right).$$

These are simply the coordinates of the small rhombicuboctahedron with the vertices of one square face removed.