Is there a typo in the formula or does my GAP-package sglppow fail?

Question

This site deals with group formulas for prime powers $p^k$ for $k\le 7$. The formula for $k=7$ seems to be wrong.

I compared the results with GAP and the formula is off by $2453$ for $p=13,17,19,23$ and $29$.

Moreover the lcm of the numbers $3,4,5,7,8,9$ is not $360$, but $2520$ and as I understood the column "modulus relative ...", the value $2520$ should be right.

The values for $p=7$ and $p=11$ coincide with the given formula.

Does anyone know the correct formula ? Does GAP give the wrong values, or is the given formula wrong ?

The formula given in the link is

$$gnu(p^7)=3p^5+12p^4+44p^3+170p^2+707p+2455$$ $$+(4p^2+44p+291)gcd(p-1,3)+(p^2+19p+135)gcd(p-1,4)$$ $$+(3p+31)gcd(p-1,5)+4gcd(p-1,7)+5gcd(p-1,8)+gcd(p-1,9)$$

What I get with GAP :

gap> LoadPackage("sglppow");
LieRing
a package for working with Lie rings 
by Serena Cicalo' and Willem de Graaf
----------------------------------------------------------------
Loading  LiePRing 1.8
by Michael Vaughan-Lee and Bettina Eick 
----------------------------------------------------------------
----------------------------------------------------------------
Loading SglPPow 1.0
by Michael Vaughan-Lee and Bettina Eick 
----------------------------------------------------------------
true
gap> gnup7:=p->3*p^5+12*p^4+44*p^3+170*p^2+707*p+2455+
> (4*p^2+44*p+291)*Gcd(p-1,3)+(p^2+19*p+135)*Gcd(p-1,4)+
> (3*p+31)*Gcd(p-1,5)+4*Gcd(p-1,7)+5*Gcd(p-1,8)+Gcd(p-1,9);
function( p ) ... end
gap> p:=Filtered([7..100],IsPrimeInt);
[ 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 
  83, 89, 97 ]
gap> r:=List( p, x -> [ x, gnup7(x), NrSmallGroups(x^7) ] );
[ [ 7, 113147, 113147 ], [ 11, 750735, 750735 ], [ 13, 1600573, 1598120 ], 
  [ 17, 5546909, 5544456 ], [ 19, 9380741, 9378288 ], 
  [ 23, 23316851, 23314398 ], [ 29, 71271069, 71268616 ], 
  [ 31, 98488755, 98486302 ], [ 37, 233043067, 233040614 ], 
  [ 41, 384847485, 384845032 ], [ 43, 485930975, 485928522 ], 
  [ 47, 751588475, 751586022 ], [ 53, 1356370173, 1356367720 ], 
  [ 59, 2299880351, 2299877898 ], [ 61, 2710679045, 2710676592 ], 
  [ 67, 4306310927, 4306308474 ], [ 71, 5734323819, 5734321366 ], 
  [ 73, 6578172579, 6578170126 ], [ 79, 9721485395, 9721482942 ], 
  [ 83, 12413061671, 12413059218 ], [ 89, 17537591045, 17537588592 ], 
  [ 97, 26866372821, 26866370368 ] ]
gap> List(r, x -> [x[2]-x[3]]);
[ [ 0 ], [ 0 ], [ 2453 ], [ 2453 ], [ 2453 ], [ 2453 ], [ 2453 ], [ 2453 ], 
  [ 2453 ], [ 2453 ], [ 2453 ], [ 2453 ], [ 2453 ], [ 2453 ], [ 2453 ], 
  [ 2453 ], [ 2453 ], [ 2453 ], [ 2453 ], [ 2453 ], [ 2453 ], [ 2453 ] ]
gap>

Answer 1

The formula on the web page you cite comes from a paper by O'Brien and Vaughan-Lee that classified the groups of order $p^7$. Web page and paper agree on the formula.

The formula gives 1600573 groups of order $13^7$ and $5546909$ groups of order $17^7$. Exactly the same numbers are given by NrSmallGroups in GAP (without special packages). I can only conclude that no discrepancy exists and that the formula is correct.

If you get reproducibly wrong numbers with a current, clean, installation of GAP, please send an error description to [email protected], including version number, system information, and the compiler you used.

Followup -- for the record as this is the only answer: It seems the `sglppow' package had a misprint in the implementation of this formula -- this bug will be corrected.