Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C214 |
| direct product | C2×#515696 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator | C23 | C25×C4 |
| Frattini, cosocle | C24 | C26 |
| prime radical | C22×#2410 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
53629 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 22 points:
(3,4,6,5)(8,10)(12,14)(15,16,19,18)(17,20,22,21);(8,10)(11,12)(13,14)(17,22)(20,21);(15,17)(16,20)(18,21)(19,22);(1,2);(7,8)(9,10)(15,17)(16,21)(18,20)(19,22);(3,5,6,4)(15,18,19,16)(17,21,22,20)256 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((3,4,6,5)(8,10)(12,14)(15,16,19,18)(17,20,22,21), (8,10)(11,12)(13,14)(17,22)(20,21), (15,17)(16,20)(18,21)(19,22), (1,2), (7,8)(9,10)(15,17)(16,21)(18,20)(19,22), (3,5,6,4)(15,18,19,16)(17,21,22,20));