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.

Statistics at a glance

Quantity	Value case $p = 2$	Value case $p = 3$	PORC function for $p \ge 5$
Total number of groups	51	67	$2p + 61 + 2\operatorname{gcd}(p-1,3) + \operatorname{gcd}(p-1,4)$
Number of abelian groups	7	7	7
Number of groups of nilpotency class exactly two	26	28	$p + 25$
Number of groups of nilpotency class exactly three	15	26	$p + 26$
Number of groups of nilpotency class exactly four (maximal class groups)	3	6	$3 + 2\operatorname{gcd}(p-1,3) + \operatorname{gcd}(p-1,4)$

Particular cases

Prime number $p$	$p^5$	Number of groups of order $p^5$	Information on groups of order $p^5$
2	32	51	groups of order 32 -- somewhat anomalous
3	243	67	groups of order 243 -- somewhat anomalous
5	3125	77	groups of order 3125
7	16807	83	groups of order 16807
11	161051	87	groups of order 161051