Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213×C4 |
| direct product | C2×#648538 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C24 | C26 |
| prime radical | C22×#2433 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
53419 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 22 points:
(3,4)(5,6)(8,9)(11,12)(13,14)(15,16,19,18)(17,21,22,20);(11,12,13,14)(15,17)(16,20)(18,21)(19,22);(1,2)(4,6)(7,8,10,9)(11,12)(13,14)(15,18,19,16)(17,21,22,20);(4,6)(7,9,10,8)(11,12)(13,14)(16,18)(17,22);(3,4)(5,6)(7,8,10,9)(15,17)(16,21)(18,20)(19,22);(15,16)(17,21)(18,19)(20,22)208 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,4)(5,6)(8,9)(11,12)(13,14)(15,16,19,18)(17,21,22,20), (11,12,13,14)(15,17)(16,20)(18,21)(19,22), (1,2)(4,6)(7,8,10,9)(11,12)(13,14)(15,18,19,16)(17,21,22,20), (4,6)(7,9,10,8)(11,12)(13,14)(16,18)(17,22), (3,4)(5,6)(7,8,10,9)(15,17)(16,21)(18,20)(19,22), (15,16)(17,21)(18,19)(20,22));