# How many groups of order $815,409=3^2\times 7^2\times 43^2$ are there?

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 $gnu(815,409)$ ?

• Which GAP command have you tried - ConstructAllGroups from GrpConst package, or NumberCFGroups from cubefree package? Jan 10 '16 at 22:53
• Can I use NumberCFGroups at once, or do I need to do something before ? I used ConstructAllGroups. Jan 10 '16 at 22:57
• 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. Jan 10 '16 at 22:59 • 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. Jan 10 '16 at 23:04 ## 1 Answer 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");
─────────────────────────────────────────────────────────────────────────────
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 --------

• The same number also arises with grpconst with the patches I recently submitted. Jan 10 '16 at 23:14