Groups of prime-sixth order

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This article is about the groups of prime-sixth order, i.e., order p^6 where p is a prime number. The cases p = 2 (see groups of order 64) and p = 3 (see groups of order 729) are somewhat different from the general case p \ge 5.

The number of groups of order p^6 for p \ge 5 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 PORC function for p \ge 5
Total number of groups 267 504 3p^2 + 39p + 344 + 24 \operatorname{gcd}(p - 1,3) + 11 \operatorname{gcd}(p-1,4) + 2 \operatorname{gcd}(p-1,5)
Number of abelian groups 11 11 11
Number of groups of nilpotency class exactly two 117 133 8p + 109
Number of groups of nilpotency class exactly three 114 282  ?
Number of groups of nilpotency class exactly four 22 71  ?
Number of groups of nilpotency class exactly five 3 7  ?

Particular cases

Note that the number of groups of order p^6 is given by the PORC function for p \ge 5.

Prime number p p^6 Number of groups Information about groups of order p^6
2 64 267 groups of order 64 -- somewhat anomalous
3 729 504 groups of order 729 -- somewhat anomalous
5 15625 684 groups of order 15625
7 117649 860 groups of order 117649
11 1771561 1192 groups of order 1771561