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I noticed that in groupprops and wikipedia there are often given tables of classifications of groups of small order. This motivated me to ask, what is the current state of research in classifying all finite groups of order $n$. I.E. what is the smallest $n$ for which we don't definitively know if we have classified all groups of that order?

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    $\begingroup$ Well 1024 is known, but not 2048. $\endgroup$
    – Gordon Royle
    Feb 19, 2017 at 6:04
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    $\begingroup$ If it isn't 2 to some small power or 3 to a smaller power, it is probably a product of both. Gerhard "Has More Than Three Primes" Paseman, 2017.02.18. $\endgroup$
    – Gerhard Paseman
    Feb 19, 2017 at 6:05
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    $\begingroup$ $1024$ is a little unclear. The number of groups of that order is known but, as far as I know, they haven't been explicitly listed, so this could depend on exactly what you mean by "classified". But the answer is certainly either $1024$ or $2048$. $\endgroup$
    – Derek Holt
    Feb 19, 2017 at 8:40
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    $\begingroup$ Not the smallest, but just a funny remark: Take a huge number $p$ for which we don't know whether $p$ is prime. Then we don't know the classification of groups of order $p$ (since there's a single one iff $p$ is prime). $\endgroup$
    – YCor
    Jan 3, 2023 at 8:40
  • $\begingroup$ @YCor There is a single group of order $n$ if and only if $n$ is squarefree, and none of its prime divisors are $1$ modulo any of the other ones. For example, there is a single group of order $15$. $\endgroup$
    – verret
    Jun 23, 2023 at 4:33

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