Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C28 · Schur multiplier (H2) = C226 |
| direct product | C22×#7708684 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C26 |
| commutator, Frattini, cosocle | C23 | C28 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1297960 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 8 generators on 24 points:
(6,8)(9,10,12,14)(11,15,16,13)(17,18,20,22)(19,23,24,21);(6,8)(9,10,12,14)(11,15,16,13)(19,24)(21,23);(9,10)(11,13)(12,14)(15,16);(3,4);(11,16)(13,15)(17,19)(18,21)(20,24)(22,23);(1,2)(3,4);(5,6)(7,8)(9,10)(11,15)(12,14)(13,16);(9,11)(10,13)(12,16)(14,15)(17,18)(19,23)(20,22)(21,24)416 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((6,8)(9,10,12,14)(11,15,16,13)(17,18,20,22)(19,23,24,21), (6,8)(9,10,12,14)(11,15,16,13)(19,24)(21,23), (9,10)(11,13)(12,14)(15,16), (3,4), (11,16)(13,15)(17,19)(18,21)(20,24)(22,23), (1,2)(3,4), (5,6)(7,8)(9,10)(11,15)(12,14)(13,16), (9,11)(10,13)(12,16)(14,15)(17,18)(19,23)(20,22)(21,24));