This shape is a "spherical pseudo-cuboctahedron". It has 24 vertices, 12 edges and 14 faces. That doesn't satisfy Euler's formula
V - E + F = 2, so it can't be a proper polyhedron - hence "pseudo-cuboctahedron".
However, if you push all the vertices onto the surface of a sphere, all the edges are spherical arcs, it sort of works.
This was requested by Jim Propp.