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 $p\geq 5$.

The number of groups of order $p^{6}$ for $p\geq 5$ is not a constant. Rather, it is given by a non-constant 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 $p\geq 5$
Total number of groups	267	504	$3p^{2}+39p+344+24\operatorname {gcd} (p-1,3)+11\operatorname {gcd} (p-1,4)+2\operatorname {gcd} (p-1,5)$
Number of abelian groups	11	11	11
Number of groups of nilpotency class exactly two	117	133	$8p+109$
Number of groups of nilpotency class exactly three	114	282	?
Number of groups of nilpotency class exactly four	22	71	?
Number of groups of nilpotency class exactly five	3	7	?

Particular cases

Note that the number of groups of order $p^{6}$ is given by the PORC function for $p\geq 5$.

Prime number $p$	$p^{6}$	Number of groups	Information about groups of order $p^{6}$
2	64	267	groups of order 64 -- somewhat anomalous
3	729	504	groups of order 729 -- somewhat anomalous
5	15625	684	groups of order 15625
7	117649	860	groups of order 117649
11	1771561	1192	groups of order 1771561