Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C216 |
| direct product | C2×#687586 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C24 | C26 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
77225 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 22 points:
(7,8)(9,10)(12,14)(16,19)(20,22);(4,6)(8,9)(17,21)(20,22);(4,6)(8,9)(11,12)(13,14)(15,16,18,19)(17,22,21,20);(1,2);(3,4,5,6)(7,9,10,8)(11,12)(13,14)(15,16)(17,20)(18,19)(21,22);(15,17)(16,20)(18,21)(19,22)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((7,8)(9,10)(12,14)(16,19)(20,22), (4,6)(8,9)(17,21)(20,22), (4,6)(8,9)(11,12)(13,14)(15,16,18,19)(17,22,21,20), (1,2), (3,4,5,6)(7,9,10,8)(11,12)(13,14)(15,16)(17,20)(18,19)(21,22), (15,17)(16,20)(18,21)(19,22));