Almost All of the First 50 Billion Groups Have Order 1024

Almost All of the First 50 Billion Groups Have Order 1024:

Here’s an incredible fact: of the 50 billion or so groups of order at
most 2000, more than 99% have order 1024. This was
announced here:

Hans Ulrich Besche, Bettina Eick, E.A. O’Brien,
The groups of order at most 2000.
Electronic Research Announcements of the American Mathematical
Society 7 (2001), 1–4.

By no coincidence, the paper was submitted in the year 2000. The real
advance was not that they had got up to order 2000, but that they had ‘developed
practical algorithms to construct or enumerate the groups of a given
order’.

I learned
this amazing nugget from a
recent
MathOverflow answer of
Ben Fairbairn.

You probably recognized that 1024=2 10. A finite group is called a
‘2-group’ if the order of every element is a power of 2, or equivalently if the
order of the group is a power of 2. So as Ben points out, what this computation
suggests is that almost every finite group is a 2-group.

Does anyone know whether there are general results making this precise? Specifically, is it true that

number of 2-groups of order  ≤Nnumber of groups of order  ≤N→1

as N→∞?

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