Groups of prime-seventh order

This article is about the groups of prime-seventh order, i.e., order $p^{7}$ where $p$ is a prime number. The cases $p=2$ (groups of order 128) and $p=3$ (groups of order 2187) are fairly different from the general case. The case $p=5$ (groups of order 78125) is somewhat anomalous, but not very different from the general case.

The number of groups of order $p^{7}$ for $p\geq 7$ 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 $p=3$	Value case $p=5$	PORC function for $p\geq 7$
Total number of groups	2328	9310	34297	$3p^{5}+12p^{4}+44p^{3}+170p^{2}+707p+2455+(4p^{2}+44p+291)\operatorname {gcd} (p-1,3)$ $+(p^{2}+19p+135)\operatorname {gcd} (p-1,4)$ $+(3p+31)\operatorname {gcd} (p-1,5)+4\operatorname {gcd} (p-1,7)+5\operatorname {gcd} (p-1,8)$$+\operatorname {gcd} (p-1,9)$
Number of abelian groups	15	15	15	15
Number of groups of nilpotency class exactly two	947	1757	7069	PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.
Number of groups of nilpotency class exactly three	1137	6050	22652	PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.
Number of groups of nilpotency class exactly four	197	1309	3274	PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.
Number of groups of nilpotency class exactly five	29	173	1188	PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.
Number of groups of nilpotency class exactly six	3	6	99	PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.

Particular cases

Prime number $p$	$p^{7}$	Number of groups	Information about groups of order $p^{7}$	Anomalous?
2	128	2328	groups of order 128	Highly
3	2187	9310	groups of order 2187	Highly
5	78125	34297	groups of order 78125	Somewhat
7	823543	113147	groups of order 823543	No
11	19487171	750735	groups of order 19487171	No

References

• The groups with order p^7 for odd prime p by E. A. O'Brien and M. R. Vaughan-Lee,  : ^{Preprint on author webpageMore info}