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OEIS shows the number of groups of order $n$ upto $2047$. The Magma-online-calculator uses a database, but already for $1024,2004,2016,...$ it cannot determine the number of groups. Maple seems to calculate the number of groups (unless it is too large), but unfortunately, I do not have access to maple.

Does anyone know an online-calculator for the number-of-group-function, or a table with the known numbers upto $10,000$ or more ?

Or alternatively, allows PARI/GP, at least in principle, to calculate the number of groups of order $n$ ? I programmed the case $n$ squarefree, but I have no clue how to manage arbitary numbers $n$.

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  • $\begingroup$ There isn't any general formula or algorithm known for computing the number of groups of a given order $n$, other than to construct all of them. Maple uses formulas for certain special cases, depending on the factorisation of $n$ (for instance, for square-free $n$, or for small powers of primes, etc.) and uses a table of known values for small $n<50000$. I expect GAP and Magma have something similar, but I don't know that PARI has that kind of functionality. $\endgroup$
    – James
    Dec 1, 2015 at 14:27
  • $\begingroup$ @James I was just curious. By the way, in the answer below, it is claimed that the number can be calculated in the case $p^2 q^2$. Do you know the formula ? $\endgroup$
    – Peter
    Dec 1, 2015 at 17:01
  • $\begingroup$ It's my understanding that some sort of formula or description is known, but I do not know it! I've never been able to get my hands on the papers where it is described. $\endgroup$
    – James
    Dec 1, 2015 at 20:33
  • $\begingroup$ For four primes, the situation is as follows. The case $p^4$ and the squarefree case are well-known. The case $p^2qr$ was done by Glenn (1906) ams.org/journals/tran/1906-007-01/S0002-9947-1906-1500737-3/… As far as I can see, the number of isomorphism type is not explicitly stated, but perhaps could be extracted with a little more work. $\endgroup$
    – verret
    Dec 2, 2015 at 6:24
  • $\begingroup$ The case $p^3q$ was done by Western (1898) plms.oxfordjournals.org/content/s1-30/1/209 His count is at the end. 15 groups when $q=2$. 6 or 19 groups when $q=3$ and $p$ odd. (For $p=2$, there are $15$.) etc... Take all of this with a grain of salt, for example, according to magma, there are $7$ groups of order $3\cdot 5^3$. Moreover, his count in the table is also off for 189 and 351. $\endgroup$
    – verret
    Dec 2, 2015 at 6:24

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For the list of $n$'s which are included in the SmallGroups library, see : https://magma.maths.usyd.edu.au/magma/handbook/text/727

There are many missing n's which could be computed, even by hand (when $n=p^2q^2$, for example), but there are definitely some numbers less than $10000$ that are out of reach.

For example, the number of groups of order $2048$ is not actually known, higher powers of $2$ even less so, see for example https://www.math.auckland.ac.nz/~obrien/research/gnu.pdf

(This is a good reference for your question in general.)

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