Groups of prime-fifth order

This article is about the groups of prime-fifth order, i.e., order $$p^5$$ where $$p$$ is an odd prime. The cases $$p = 2$$ (see groups of order 32) and $$p = 3$$ (see groups of order 243) are somewhat different from the general case $$p \ge 5$$.

$$p^5$$ is the smallest prime power for which the number of groups of that order is not eventually constant, but rather, is given by a nonconstant PORC function in keeping with Higman's PORC conjecture.