# 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