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Groups of order $2^n$ have most non-isomorphic types and are notoriously hard to classify. For example, order $32$ has $51$ non-isomorphic groups, and order $64$ has $267$ ones. In general group theory texts, only groups of order up to $15$ are (completely) classified. Even the order of $16$ is not available for a full classification.

On this website, all groups up to order $500$ are classified with given names. My question is, are groups of $64$ or $128$ or higher classified manually through presentations or by programming (computational group theory)?

Eugene Zhang
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    It becomes extremely difficult. See this for example: https://math.stackexchange.com/questions/241369/more-than-99-of-groups-of-order-less-than-2000-are-of-order-1024 . The number of groups of order $2048$ is, as far as I know, still unknown. ā€“ pancini Feb 28 '24 at 20:32

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Order 64 was done by hand still -- Hall, Marshall, Jr.; Senior, James K. The groups of order $2^n$ ($n\le 6$). Macmillan, 1964. Order 128 is by computer, but would have been possible by hand, see also O'Brien, Vaughan-Lee The groups with order $p^7$ for odd prime $p$. J. Algebra 292 (2005), 243ā€“258

ahulpke
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It is known that the number of isomorphism classes of groups of order $p^n$ ($p$ prime) grows as $p^{{\frac{2}{27}n^3} +O(n^{8/3})}$ (Charles Sims, Enumerating p-groups, Proc. London Math. Soc., Series 3, 15 (1965), 151-166).

Because of this exponential growth, is it believed that almost all finite groups are $2$-groups: the fraction of isomorphism classes of $2$-groups among isomorphism classes of groups of order at most $n$ is thought to tend to $1$ as $n$ tends to infinity. If you look at the table in this paper of Conway, Dietrich and O'Brien, you can see that of the 49 910 529 484 different groups of order at most $2000$, 49 487 367 289, or just over 99%, are the $2$-groups of order 1024.

The first to create some order in the plethora of groups of prime-power order was Philip Hall (The classification of prime-power groups, J. Reine Angew. Math. 182 (1940),130-141). He observed that the notion of isomorphism of groups is really too strong to give rise to a satisfactory classification and that it had to be replaced by a weaker equivalence relation. Subsequently he discovered a suitable equivalence relation and called it isoclinism of groups. It is this classification principle that underlies the mentioned monograph of M. Hall and J. K. Senior on the classification of $2$-groups of order at most $64$ (see their first DEFINITION, where isoclinism class is not mentioned but rather family). Another strategy for classification is looking at coclasses: the coclass of a finite $p$-group of order $p^n$ and nilpotency class $c$ is $nāˆ’c$.

Yet another approach is that of Marcus du Sautoy (1999): he constructs certain zeta-functions and uses $p$-adic integrals to capture the number of finite $p$-groups, see this paper, or Zeta functions of Groups: the Quest for Order versus the Flight from Ennui, which appeared in Groups St Andrews 2001, LMS Lecture Note Series 304 (2001), 150-189.

Nicky Hekster
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  • Note that while $p^{{\frac{2}{27}n^3}+O(n^{8/3})}$ is (super)exponential in $n$, it is quasipolynomial in $p^n$, which is the order of the group and the more relevant quantity if you are going to be comparing to groups of arbitrary order. ā€“ verret Feb 29 '24 at 19:32