Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C218×C4 |
| direct product | C22×#35292 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C23 | C27 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
104669 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 7 generators on 24 points:
(6,8)(9,10,13,11)(12,15,16,14)(17,18,21,24)(19,22,23,20);(5,6)(7,8)(10,11)(14,15);(9,10,13,11)(12,14,16,15)(17,19,21,23)(18,22,24,20);(9,11,13,10)(12,15,16,14)(17,18)(19,22)(20,23)(21,24);(9,12,13,16)(10,14,11,15)(17,20,21,22)(18,23,24,19);(3,4);(1,2)(3,4)256 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((6,8)(9,10,13,11)(12,15,16,14)(17,18,21,24)(19,22,23,20), (5,6)(7,8)(10,11)(14,15), (9,10,13,11)(12,14,16,15)(17,19,21,23)(18,22,24,20), (9,11,13,10)(12,15,16,14)(17,18)(19,22)(20,23)(21,24), (9,12,13,16)(10,14,11,15)(17,20,21,22)(18,23,24,19), (3,4), (1,2)(3,4));