... I'd venture to suggest - see references:

http://mathaware.org/mam/00/master/essays/SciAm/SA04.html
"Nevertheless, the number of vertexes, edges, faces and hyperfaces (ordinary cubes) that make up the hypercube can readily be counted. The number of vertexes is just the number of vertexes in the initial cube plus the number in the final cube, or 16. Each of the eight vertexes in the initial cube is joined by an edge to one of the eight vertexes in the final cube, and there are also 12 edges in each of the two cubes. Hence there are 9 + 12 + 12, or 32, edges in the hypercube. One can also show that the hypercube has 24 ordinary faces and eight hyperfaces."

http://mathworld.wolfram.com/Cube5-Compound.html
"The compound can be generated by starting with an initial cube centered at the origin and oriented along the axes, then adding four more cubes obtained from the initial cube by rotations through angles..."