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Probably, the number of groups of order $18,480$ can only be determined with GAP. But I may be wrong, so the question should not be understood to be purely computational. A better lower bound than my mentioned $1,005$ or a good upper bound would also be welcome.

I tried to determine the number of groups of order $18,480$ with GAP.

My GAP version once again failed because of a memory overflow. The final output :

#I     Iso: test isomorphism on groups of size 18480
gap: cannot extend the workspace any more!
gap: Press <Enter> to end program

What is $gnu(18,480)$ ? Please approve with ForAll(x,IsGroup) , that the groups are actually non-isomorphic.

What is the best upper bound, which can be determined without great effort ?

The calculation for $9,240$ worked well and gives a lower bound for the desired value.

9240:779:[ [ 2, 3 ], [ 3, 1 ], [ 5, 1 ], [ 7, 1 ], [ 11, 1 ] ]:true

A better lower bound is $gnu(1,680)=1,005$.

Since the factorization of $\ \ 18,480=2^4\cdot3\cdot5\cdot7\cdot11\ \ $ has no large prime-power, there might be a method to determine the number of groups even without electronic help. Is someone has an idea, he/she is invited to show the method and the result.

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  • $\begingroup$ How long did it take to run till the memory overflow? $\endgroup$ – Alexander Konovalov May 1 '16 at 17:12
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    $\begingroup$ @AlexanderKonovalov I would say 1-2 hours. I also have problems with the time-command. It should display the duration of the last calculation in milliseconds. Once I got a value much too low, and once even a negative value... $\endgroup$ – Peter May 1 '16 at 17:48
  • $\begingroup$ @AlexanderKonovalov What do you think ? Is there any possibility to determine the number of gnu's of a given order WITHOUT storing the constructed groups ? $\endgroup$ – Peter May 1 '16 at 17:55
  • $\begingroup$ I suggest to always add even a rough estimate of time: knowing just whether it is measured in seconds/minutes/hours/days is helpful to the reader to decide when and where to try to run this calculation. The problem with time overflow after several hours of computation was reported before. It may be Windows Vista specific. $\endgroup$ – Alexander Konovalov May 2 '16 at 12:26
  • $\begingroup$ The memory overflow could be caused by the isomorphism check and not by storing all groups detected so far, though those of course contributed to memory usage. Regarding the other question, probably the algorithm does need to keep all groups discovered so far in memory, and can work in "small batches" (which could be also parallelised), but that requires reengineering of ConstructAllGroups. $\endgroup$ – Alexander Konovalov May 2 '16 at 12:31
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$5397$ isomorphism classes of groups. About 5 hours, under 1 GB in (development version of) GAP.

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  • $\begingroup$ Thanks. I've checked this with GAP 4.8.3, in 10 hours (perhaps on a slower machine). $\endgroup$ – Alexander Konovalov May 2 '16 at 8:02
  • $\begingroup$ Now added to github.com/alex-konovalov/gnu/blob/master/gnudata.g - there are already 10 precomputed values of $gnu(n)$ stored there. $\endgroup$ – Alexander Konovalov May 2 '16 at 12:22
  • $\begingroup$ @AlexanderKonovalov Did you check with ForAll(x,IsGroup) ? $\endgroup$ – Peter May 2 '16 at 13:10
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    $\begingroup$ @Peter yes, there were no lists returned $\endgroup$ – Alexander Konovalov May 2 '16 at 13:16

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