Groups of order 8

Statistics at a glance
Since $$8 = 2^3$$ is a prime power, and prime power order implies nilpotent, all groups of this order are nilpotent groups.

Presentations
Below are the power-commutator presentations for groups of order 8.

Functions taking values between 0 and 3
Here are some measures of central tendency (averages) and deviation measures:

Assuming equal weighting on all groups:

Same, with rows and columns interchanged:

Here are the correlations between these various arithmetic functions across the groups of order 8:

Arithmetic function values of a counting nature
Here is the same table, with rows and columns interchanged:

Up to isoclinism
The equivalence classes up to being isoclinic were classified by Hall and Senior, and we call them Hall-Senior families.

Up to Hall-Senior genus
Up to the relation of groups having the same Hall-Senior genus, there are four equivalence classes:

Up to isologism for higher class
Since all the groups of order 8 has class at most two, we have a unique equivalence class under isologism for any class equal to or more than two.

Up to isologism for elementary abelian groups
Each of the abelian groups is in a different equivalence class under the equivalence relation of being isologic with respect to elementary abelian 2-groups. The two non-abelian groups are isologic to each other with respect to the variety of elementary abelian 2-groups.

Order statistics
Here are the statistics for a particular order.

Here are the number of $$n^{th}$$ root statistics. The number of $$n^{th}$$ roots equals the number of elements whose order divides $$n$$.

Equivalence classes
No two of the groups of order 8 are order statistics-equivalent, and hence no two of them are 1-isomorphic.