Classification of groups of prime-fourth order

Statement

Let $p$ be a prime number. Then, the groups of order $p^{4}$ can be classified as follows, with slightly different classifications for the cases $p=2$, $p=3$, and $p\geq 5$. The case $p=2$ is different even in so far as the number of possible groups is concerned. The cases $p=3$ and $p\geq 5$ have minor differences with each other.

Group property	Number of isomorphism classes of groups case $p=2$, so order $16$	Number of isomorphism classes of groups case odd prime
abelian group	5	5
nilpotency class exactly two, i.e., class two group that is non-abelian	6	6
nilpotency class exactly three	3	4
Total	14	15

The five abelian groups

The nature and classification of the five abelian groups of order $p^{4}$ is the same for both the $p=2$ and odd $p$ cases; the abelian groups are classified by the set of unordered integer partitions of the number 4.

Partition of 4	Corresponding abelian group (in general)	Corresponding abelian group case $p=2$	GAP ID (2nd part) case $p=2$	GAP ID (2nd part) case odd $p$
4	cyclic group of prime-fourth order	cyclic group:Z16	1	1
3 + 1	direct product of cyclic group of prime-cube order and cyclic group of prime order	direct product of Z8 and Z2	5	5
2 + 2	direct product of cyclic group of prime-square order and cyclic group of prime-square order	direct product of Z4 and Z4	2	2
2 + 1 + 1	direct product of cyclic group of prime-square order and elementary abelian group of prime-square order	direct product of Z4 and V4	10	11
1 + 1 + 1 + 1	elementary abelian group of prime-fourth order	elementary abelian group:E16	14	15