Groups of prime-sixth order

This article is about the groups of prime-sixth order, i.e., order $$p^6$$ where $$p$$ is a prime number. The cases $$p = 2$$ (see groups of order 64) and $$p = 3$$ (see groups of order 729) are somewhat different from the general case $$p \ge 5$$.

The number of groups of order $$p^6$$ for $$p \ge 5$$ is not a constant. Rather, it is given by a non-constant PORC function in keeping with Higman's PORC conjecture.

Particular cases
Note that the number of groups of order $$p^6$$ is given by the PORC function for $$p \ge 5$$.