Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C219 |
| direct product | C22×#45357 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C23 | C27 |
| prime radical | C23×#127 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
86997 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)(9,10,14,16)(11,15,12,13);(5,6)(7,8)(9,11,14,12)(10,13,16,15)(17,18,20,22)(19,21,24,23);(9,11,14,12)(10,13,16,15)(17,18)(19,21)(20,22)(23,24);(9,12,14,11)(10,15,16,13)(17,19,20,24)(18,21,22,23);(9,13,14,15)(10,12,16,11)(17,18)(19,23)(20,22)(21,24);(3,4);(1,2)(3,4)232 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)(9,10,14,16)(11,15,12,13), (5,6)(7,8)(9,11,14,12)(10,13,16,15)(17,18,20,22)(19,21,24,23), (9,11,14,12)(10,13,16,15)(17,18)(19,21)(20,22)(23,24), (9,12,14,11)(10,15,16,13)(17,19,20,24)(18,21,22,23), (9,13,14,15)(10,12,16,11)(17,18)(19,23)(20,22)(21,24), (3,4), (1,2)(3,4));