Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C28 · Schur multiplier (H2) = C226 |
| direct product | C22×#7763449 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C26 |
| commutator, Frattini, cosocle | C23 | C28 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
At least 9690 conjugacy classes of subgroups; only the 445480 normal subgroups are shown.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 8 generators on 24 points:
(5,6,8,7)(10,13)(11,15)(17,18)(19,21)(20,23)(22,24);(5,6,8,7)(10,13)(14,16)(17,19,20,24)(18,21,23,22);(5,6)(7,8)(9,10,12,13)(11,14,15,16)(19,24)(21,22);(3,4);(6,7)(17,18)(19,21)(20,23)(22,24);(1,2)(3,4);(9,10)(11,16)(12,13)(14,15)(17,18)(19,21)(20,23)(22,24);(5,7,8,6)(9,11,12,15)(10,14,13,16)(17,19)(18,22)(20,24)(21,23)392 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((5,6,8,7)(10,13)(11,15)(17,18)(19,21)(20,23)(22,24), (5,6,8,7)(10,13)(14,16)(17,19,20,24)(18,21,23,22), (5,6)(7,8)(9,10,12,13)(11,14,15,16)(19,24)(21,22), (3,4), (6,7)(17,18)(19,21)(20,23)(22,24), (1,2)(3,4), (9,10)(11,16)(12,13)(14,15)(17,18)(19,21)(20,23)(22,24), (5,7,8,6)(9,11,12,15)(10,14,13,16)(17,19)(18,22)(20,24)(21,23));