Computational group theory
GAP1024210
Every group of order 1024, written down one by one.
There are 49,487,367,289 groups of order 1024. David Burrell counted them in 2022, but most of them had never been written down. GAP1024 assembles all of them into one library: each group appears exactly once, as an explicit presentation, and every family is checked against Burrell's count.
The library
Two numbers sort the groups. The rank of a group G is its minimal number of generators, so that |G : Φ(G)| = 2d. The p-class is the length of the lower exponent-2 central series, which runs from G down to 1 with each term [P, G]P2 of the one before. The groups of p-class 2, those whose Frattini subgroup is central and elementary abelian, form cells (K, E): rank K and |Φ(G)| = 2E, with K + E = 10. A third number, the nilpotency class, is the length of the lower central series G > [G, G] > [G, G, G] > … > 1; it is at most the p-class.
Where the groups are
Bars are to scale. One cell, (6, 4), holds 96.4% of all groups, so each bar below magnifies the hatched remainder of the bar above it. Hover over a block, or tap it, for details.
By p-class and rank
Show the numbers as a table
Every count is checked against Burrell's. For p-class 2, two independent Cauchy–Frobenius programs reproduce his numbers too: the groups in cell (K, E) correspond to the orbits of GL(K, 2) on E-dimensional spaces of quadratic forms on F2K. The counts by rank and p-class outside p-class 2 come from Burrell's GRPS1024 and agree with the groups in the library.
Group explorer
Paste a record, pick an example, or draw a random group. The page computes the invariants and presentations from the record alone, exactly and in your browser; nothing is sent anywhere. Records cover the groups of p-class 2.
Browse and filter
Three cells of p-class 2 are small enough to list in full here: 6,512 groups. Filter them by their invariants, open any one in the explorer, or copy GAP code that builds the whole selection.
How a record works
A record K E f1 … fE lists the cell and E integers. Write the generators of G modulo Φ(G) as g1, …, gK and a basis of Φ(G) as z1, …, zE. Bit i of fl says whether zl occurs in the square gi+12. The higher bits do the same for the commutators [gi, gj], with the pairs taken in the order (1, 2), (1, 3), …, (K − 1, K).
In group terms, each fl is a quadratic form on F2K. Many records describe the same group, since a change of generators changes the forms, and the library stores exactly one record per group. In the cells of rank at most 7 it is the canonical record, so two groups there are isomorphic exactly when their canonical records agree; for the cells (8, 2) and (9, 1) it is a fixed representative. Groups of p-class 1 or at least 3 use GAP pc codes instead, the format of GAP's SmallGroups library, and load with PcGroupCode(code, 1024).
Try it: the groups of order 8
One generator z of Φ(G), two generators g1, g2. Switch the three bits of f.
References
David Burrell, On the number of groups of order 1024, Communications in Algebra 50(6) (2022), 2408–2410. doi:10.1080/00927872.2021.2006680
David Burrell, GRPS1024, a GAP package (Artistic License 2.0). The groups of p-class 1 and 3 to 10 in the library, and their counts by rank, come from its data.
The GAP Group, GAP — Groups, Algorithms, and Programming. gap-system.org
The code, the library and a paper on it will be released soon.