Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213×C4 |
| direct product | C2×#2109191 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C24 | C26 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
41237 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)(12,14)(15,16,18,20)(17,21,22,19);(3,4)(5,6)(8,10)(11,12)(13,14)(17,22)(19,21);(3,4)(5,6)(8,10)(15,16,18,20)(17,21,22,19);(1,2);(4,6)(7,8,9,10)(15,16)(17,19)(18,20)(21,22);(3,4)(5,6)(15,17)(16,19)(18,22)(20,21)160 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)(12,14)(15,16,18,20)(17,21,22,19), (3,4)(5,6)(8,10)(11,12)(13,14)(17,22)(19,21), (3,4)(5,6)(8,10)(15,16,18,20)(17,21,22,19), (1,2), (4,6)(7,8,9,10)(15,16)(17,19)(18,20)(21,22), (3,4)(5,6)(15,17)(16,19)(18,22)(20,21));