Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C219 |
| direct product | C22×#52032 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C23 | C27 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
93337 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 7 generators on 24 points:
(6,8)(9,10,13,16)(11,12,15,14);(5,6)(7,8)(9,11)(10,14)(12,16)(13,15)(17,18)(19,20)(21,24)(22,23);(9,11,13,15)(10,14,16,12)(17,19)(18,22)(20,24)(21,23);(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24);(9,12)(10,15)(11,16)(13,14)(17,20)(18,23)(19,24)(21,22);(3,4);(1,2)(3,4)208 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)(9,10,13,16)(11,12,15,14), (5,6)(7,8)(9,11)(10,14)(12,16)(13,15)(17,18)(19,20)(21,24)(22,23), (9,11,13,15)(10,14,16,12)(17,19)(18,22)(20,24)(21,23), (9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24), (9,12)(10,15)(11,16)(13,14)(17,20)(18,23)(19,24)(21,22), (3,4), (1,2)(3,4));