Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C28 · Schur multiplier (H2) = C227 |
| direct product | C23×#36123 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C26 | C25 |
| commutator, Frattini, cosocle | C23 | C28 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C26 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
2140216 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 8 generators on 22 points:
(7,8,10,9)(12,13)(15,16,18,20)(17,21,22,19);(11,12,14,13)(15,16)(17,19)(18,20)(21,22);(15,16)(17,19)(18,20)(21,22);(3,4);(5,6);(8,9)(15,16)(17,21)(18,20)(19,22);(1,2)(3,4)(5,6);(7,9,10,8)(11,13,14,12)(15,17)(16,19)(18,22)(20,21)560 conjugacy classes. The character table is too large to display.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((7,8,10,9)(12,13)(15,16,18,20)(17,21,22,19), (11,12,14,13)(15,16)(17,19)(18,20)(21,22), (15,16)(17,19)(18,20)(21,22), (3,4), (5,6), (8,9)(15,16)(17,21)(18,20)(19,22), (1,2)(3,4)(5,6), (7,9,10,8)(11,13,14,12)(15,17)(16,19)(18,22)(20,21));