Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C221 |
| direct product | C22×#1006038 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C26 | C25 |
| commutator, Frattini, cosocle | C24 | C27 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C26 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
888304 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 7 generators on 24 points:
(9,10,12,11)(13,14,15,16)(17,18)(19,20)(21,22)(23,24);(3,4);(10,11)(13,14)(15,16)(17,19,22,24)(18,23,21,20);(1,2)(3,4);(10,11)(18,21)(19,24);(6,8)(17,20)(18,19)(21,24)(22,23);(5,6)(7,8)(9,11,12,10)(17,21,22,18)(19,23,24,20)344 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((9,10,12,11)(13,14,15,16)(17,18)(19,20)(21,22)(23,24), (3,4), (10,11)(13,14)(15,16)(17,19,22,24)(18,23,21,20), (1,2)(3,4), (10,11)(18,21)(19,24), (6,8)(17,20)(18,19)(21,24)(22,23), (5,6)(7,8)(9,11,12,10)(17,21,22,18)(19,23,24,20));