# 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?
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You might start by explaining what exactly we are looking at. And then maybe try to give some hint of what you might mean by "resemble". – Marc van Leeuwen Mar 24 '13 at 6:37
We're looking at all the Cayley tables of groups from GAP. I chose the ordering from GAP, so we're looking at SmallGroup(128,n) for n from 1 to 2328. The coloring is linear so element 1 gets mapped to black and element 128 mapped to white. – deoxygerbe Mar 24 '13 at 6:46
It's clear what's being asked. OP sees a weird pattern in the Cayley table and he wants to know what's up with that. Perfectly fine question – Alexander Gruber Mar 29 '13 at 4:02

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.