The title refers not to numbers of the form p 3, p prime, but to geometric cubes with p 3 subcubes.
Such cubes are natural models for the finite vector spaces acted upon by general linear groups viewed as permutation groups of degree (not order ) p 3.
For the case p =2, see The Eightfold Cube.
For the case p =3, see the "External links" section of the Nov. 30, 2009, version of Wikipedia article "General Linear Group." (That is the version just prior to the Dec. 14, 2009, revision by anonymous user "Greenfernglade.")
For symmetries of group actions for larger primes, see the related 1985 remark* on two -dimensional linear groups—
"Actions of GL(2,p ) on a p ×p coordinate-array
have the same sorts of symmetries,
where p is any odd prime."