Groups of prime-cube order

This article is about the groups of prime-cube order for an odd prime number, i.e., the groups of order $$p^3$$ where $$p$$ is an odd prime. The special case $$p = 2$$ is somewhat different -- see groups of order 8 for a summary of information on these groups.

The list
The list below is valid for odd primes. The list is somewhat different for $$p = 2$$; see groups of order 8.

Functions taking values between 0 and 3
These arithmetic function values are the same for all $$p \ne 2$$ for the corresponding groups. For $$p = 2$$, the behavior for the abelian groups is exactly the same, but the two non-abelian groups behave a little differently.

Here now is the same table with various measures of averages and deviations:

Same, with rows and columns interchanged:

Here are the correlations between the arithmetic function values for the groups of order $$p^3$$:

Arithmetic function values of a counting nature
Same, with rows and columns interchanged:

Order statistics
Here are the cumulative order statistics, where the number of $$n^{th}$$ roots is the number of elements whose order divides $$n$$.

Equivalence classes
Up to order statistics-equivalence, there are three equivalence classes. Moreover, these are the same as the eqiuvalence classes up to 1-isomorphism. A deeper explanation of this is that all the group in a given equivalence class actually have the same additive group of their respective Lazard Lie rings.