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The number of groups of order $2048$ is unknown.

What are the tightest known bounds (lower and upper bounds : I am interested in both) for the number of groups of order $2048$ ?

I know the asymptotic formula for $p^k$, but I do not think that it gives a useful bound for $p^k=2048$. Somewhere, I read that a subset of the groups (but I do not remember what kind of subset) was calculated to obtain a lower bound.

Can it be estimated how long it will take to determine the number ?

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  • $\begingroup$ Should I post this question also in the math-overflow-forum ? $\endgroup$ – Peter Dec 2 '15 at 15:30
  • $\begingroup$ The exact number is not known: groupprops.subwiki.org/wiki/Groups_of_order_2048 $\endgroup$ – lhf Dec 2 '15 at 15:34
  • $\begingroup$ @Ihf This is the reason, I am asking for bounds. $\endgroup$ – Peter Dec 2 '15 at 15:37
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    $\begingroup$ It would be an appropriate question for mathoverflow, but the convention is that you should not post to both sites simultaneously, so you should wait a few hours before posting it there. You could also try e-mailing Eamon O'Brien and askig him, $\endgroup$ – Derek Holt Dec 2 '15 at 16:30
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According to the second paragraph of https://www.math.auckland.ac.nz/~obrien/research/gnu.pdf (which I posted in another of your questions yesterday)

"[The number of groups of order 2048] is still not precisely known, but it strictly exceeds 1774274116992170, which is the exact number of groups of order 2048 that have exponent-2 class 2, and can confidently be expected to agree with that number in its first 3 digits. "

This is probably the best answer you can get easily.

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  • $\begingroup$ 8843740 of the 10494213 groups of order 512 (about 84%) have nilpotency class $2$ (I didn't calculate the exponent $2$-class - I can do that). $\endgroup$ – Derek Holt Dec 3 '15 at 9:37
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I believe the subset of groups you're thinking of is the set of $p$-class 2 groups of order $p^n$. Higman uses this to obtain his lower bound which is $$ p^{\frac{2}{27}(n^3-6n^2)}.$$ Thus, a lower bound on the number of groups of order $2^{11}$ is $2^{\frac{1210}{27}}\approx 3\times 10^{13}$.

In that same paper, Higman computes an easy upper bound of $$p^{\frac{1}{6}(n^3-n)},$$ which gives us an upper bound of $2^{220}\approx 10^{66}$.

The upper bound here is not the best known (even Higman gives a slightly better asymptotic bound in the same paper), but I think the lower bound might be the best we have.

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    $\begingroup$ It can't be the sharpest because not all such groups have class $2$. $\endgroup$ – Derek Holt Dec 2 '15 at 23:01
  • $\begingroup$ Yes of course. I hope I have clarified what I meant to say. Thank you! $\endgroup$ – JMag Dec 2 '15 at 23:54

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