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

Question

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 $gnu(815,409)$ ?

Answer 1

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, [email protected] -------- 
Loading Cubefree 1.15 ... 
true
gap> NumberCFGroups(815409);
415
gap> time;
61078