why do most finite groups of order 128 resemble (at a distance) the elementary abelian group?

As a result of this previous question, I made the following video:

Cayley Tables of All Groups of Order 128, and what is striking is that most of them, if you squint, kind of resemble the elementary abelian group (gap, SmallGroups(128,2328)):

• Is there a group-theoretical reason that this might be the case?
• Is there a measure of how far one of these groups is away from being the elementary abelian group?

Solution 1:

Firstly, cool movie!

I'm guessing here a bit since I don't know the precise details of how you created the movie and the precise ordering you chose (which might cluster things together in such a way as to 'hide' many cases that look differently). $128$ is very special and allows very little room for variation in the group structure. You will only see interesting subgroups of order $2,4,8, 16, 32, 64$ with the huge majority having subgroups of order $8$, $16$ (I believe). The elementary abelian group is not that far from a group where all elements have order at most, say $8$, so most groups will appear very similar to the elementary abelian group.

Again, this is mostly a guess. You might want to play with sizes that are not a multiple of just a single prime to see if that significantly alters the picture.