Groups of prime-fourth order

This article is about the groups of prime-fourth order for an odd prime number, i.e., the groups of order $p^{4}$ where $p$ is an odd prime. The special case $p=2$ is somewhat different -- see groups of order 16 for a summary of information on these groups.

Among the odd primes $p$, the case $p=3$ is slightly different from the other primes.

Statistics at a glance

Group counts

Quantity	Value case $p=2$	Value case $p=3$	Value case $p\geq 5$	Explanation
Total number of groups	14	15	15	See classification of groups of order 16, classification of groups of prime-fourth order for odd prime
Number of abelian groups	5	5	5	See classification of finite abelian groups and structure theorem for finitely generated abelian groups.. In this case, the number of unordered integer partitions of 4 equals 5.
Number of groups of nilpotency class exactly two	6	6	6
Number of groups of nilpotency class exactly three	3	4	4

Particular cases

Prime $p$	$p^{4}$	Number of groups	Information on groups of order $p^{4}$
2	16	14	groups of order 16 -- behaves quite differently from the others.
3	81	15	groups of order 81 -- behaves somewhat differently from the others.
5	625	15	groups of order 625
7	2401	15	groups of order 2401
11	14641	15	groups of order 14641