Groups of prime-seventh order

This article is about the groups of prime-seventh order, i.e., order $$p^7$$ where $$p$$ is a prime number. The cases $$p = 2$$ (groups of order 128) and $$p = 3$$ (groups of order 2187) are fairly different from the general case. The case $$p = 5$$ (groups of order 78125) is somewhat anomalous, but not very different from the general case.

The number of groups of order $$p^7$$ for $$p \ge 7$$ is not a constant. Rather, it is given by a non-constant PORC function in keeping with Higman's PORC conjecture.