Groups of prime-fifth order

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This article is about the groups of prime-fifth order, i.e., order p^5 where p is an odd prime. The cases p = 2 (see groups of order 32) and p = 3 (see groups of order 243) are somewhat different from the general case p \ge 5.

p^5 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 p = 2 Value case p = 3 PORC function for p \ge 5
Total number of groups 51 67 2p + 61 + 2\operatorname{gcd}(p-1,3) + \operatorname{gcd}(p-1,4)
Number of abelian groups 7 7 7
Number of groups of nilpotency class exactly two 26 28 p + 25
Number of groups of nilpotency class exactly three 15 26 p + 26
Number of groups of nilpotency class exactly four (maximal class groups) 3 6 3 + 2\operatorname{gcd}(p-1,3) + \operatorname{gcd}(p-1,4)

Particular cases

Prime number p p^5 Number of groups of order p^5 Information on groups of order p^5
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