I aborted GAP after some hours. I wanted to approve my conjecture that $gnu(n)<n$ for all cubefree numbers $n>1$, where $gnu(n)$ is the number of groups of order $n$, but the case $p^2\times q^2\times r^2$ seems to be already complicated.

Does anyone know whether $gnu(815,409)<815,409$ holds, or even the value of $gnu(815,409)$ ?

  • $\begingroup$ Which GAP command have you tried - ConstructAllGroups from GrpConst package, or NumberCFGroups from cubefree package? $\endgroup$ Jan 10 '16 at 22:53
  • $\begingroup$ Can I use NumberCFGroups at once, or do I need to do something before ? I used ConstructAllGroups. $\endgroup$
    – Peter
    Jan 10 '16 at 22:57
  • $\begingroup$ Also, I suggest to look at getting account at SageMathCloud where you can leave some calculations running for a longer time (free accounts are available, you will get more power with paid accounts starting from IIRC US$ 7 per month). Also, there is MEDICIS though I haven't used it for a while, but may be worth trying. $\endgroup$ Jan 10 '16 at 22:59
  • $\begingroup$ I prefer offline-solutions. If they do not work, it is OK. I am not a friend of programs accessing to the internet, and I avoid paid services in the internet in principle. It seems too dangerous for me. I have great interest in group theory at the moment, but it is after all limited. $\endgroup$
    – Peter
    Jan 10 '16 at 23:04

Short answer: $415$.

In more details: for cube-free orders, one should use the Cubefree package by Heiko Dietrich. It takes about a minute to get the answer, which is $415$, using the beta-version of GAP 4.8 (namely, GAP 4.8.1) with most recent GrpConst and Cubefree:

gap> LoadPackage("cubefree");
Loading  GrpConst 2.5 (Constructing the Groups of a Given Order)
by Hans Ulrich Besche and
   Bettina Eick (http://www.icm.tu-bs.de/~beick).
Homepage: http://www.icm.tu-bs.de/~beick/so.html

   - Construction Algorithm for Cubefree Groups, 1.11 - 
   ------- Heiko Dietrich, H.Dietrich@tu-bs.de -------- 
Loading Cubefree 1.15 ... 
gap> NumberCFGroups(815409);
gap> time;
  • $\begingroup$ The same number also arises with grpconst with the patches I recently submitted. $\endgroup$
    – ahulpke
    Jan 10 '16 at 23:14
  • $\begingroup$ If you will try GAP 4.7, you will get an error - this will be fixed in GAP 4.8 (that's why I was using its beta-version). $\endgroup$ Jan 10 '16 at 23:16
  • $\begingroup$ @ahulpke great, good to have this cross-check! $\endgroup$ Jan 10 '16 at 23:18
  • $\begingroup$ Would it be sufficient to add some files to my current GAP-version ? $\endgroup$
    – Peter
    Jan 10 '16 at 23:23
  • 3
    $\begingroup$ Why bad news? Is installing a new gap version such a hurdle? If so, that is bad news to us, and I'd love to hear why exactly it is such a problem, so that perhaps we can make it easier for you in the future. Any replies most welcome (also fell free to email me, or email the GAP support list.) Thank you! $\endgroup$
    – Max Horn
    Jan 11 '16 at 7:52

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