Groups of prime-fifth order
From Groupprops
This article is about the groups of prime-fifth order, i.e., order where
is an odd prime. The cases
(see groups of order 32) and
(see groups of order 243) are somewhat different from the general case
.
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 ![]() |
Value case ![]() |
PORC function for ![]() |
---|---|---|---|
Total number of groups | 51 | 67 | ![]() |
Number of abelian groups | 7 | 7 | 7 |
Number of groups of nilpotency class exactly two | 26 | 28 | ![]() |
Number of groups of nilpotency class exactly three | 15 | 26 | ![]() |
Number of groups of nilpotency class exactly four (maximal class groups) | 3 | 6 | ![]() |
Particular cases
Prime number ![]() |
![]() |
Number of groups of order ![]() |
Information on groups of order ![]() |
---|---|---|---|
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 |