Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212×C42 |
| direct product | C2×#625934 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C24 | C26 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
60313 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 22 points:
(4,5)(7,8)(9,10);(4,5)(8,9)(12,14)(15,16,18,20)(17,19,22,21);(15,16)(17,19)(18,20)(21,22);(1,2);(3,4,6,5)(7,8,10,9)(11,12,13,14)(15,16)(17,21)(18,20)(19,22);(3,5,6,4)(7,9,10,8)(15,17)(16,19)(18,22)(20,21)184 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((4,5)(7,8)(9,10), (4,5)(8,9)(12,14)(15,16,18,20)(17,19,22,21), (15,16)(17,19)(18,20)(21,22), (1,2), (3,4,6,5)(7,8,10,9)(11,12,13,14)(15,16)(17,21)(18,20)(19,22), (3,5,6,4)(7,9,10,8)(15,17)(16,19)(18,22)(20,21));