Classification of groups of prime-sixth order

Statement

Let $p$ be a prime number. Then, the groups of order $p^{6}$ can be classified as follows, with slightly different classifications for the cases $p=2$, $p=3$, and $p\geq 5$.

The number of groups

• There are $267$ groups of order $2^{6}=64$, see groups of order 64
• There are $504$ groups of order $3^{6}=729$, see groups of order 729
• There are $3p^{2}+39p+344+24\mathrm {gcd} (p-1,3)+11\mathrm {gcd} (p-1,4)+2\mathrm {gcd} (p-1,5)$ of order $p^{6}$ for $p\geq 5$.

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.