Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C219 |
| direct product | C22×#42952 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C23 | C27 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
82277 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 7 generators on 24 points:
(5,6)(7,8)(17,18)(19,23)(20,22)(21,24);(5,6)(7,8)(9,10)(11,14)(12,15)(13,16)(17,19)(18,21)(20,24)(22,23);(5,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,18,20,22)(19,23,24,21);(6,7)(9,11)(10,14)(12,16)(13,15)(17,18)(19,21)(20,22)(23,24);(5,7,8,6)(9,10)(11,13)(12,15)(14,16)(17,19)(18,21)(20,24)(22,23);(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((5,6)(7,8)(17,18)(19,23)(20,22)(21,24), (5,6)(7,8)(9,10)(11,14)(12,15)(13,16)(17,19)(18,21)(20,24)(22,23), (5,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,18,20,22)(19,23,24,21), (6,7)(9,11)(10,14)(12,16)(13,15)(17,18)(19,21)(20,22)(23,24), (5,7,8,6)(9,10)(11,13)(12,15)(14,16)(17,19)(18,21)(20,24)(22,23), (3,4), (1,2)(3,4));