Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C219 |
| direct product | C22×#36469 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C23 | C27 |
| prime radical | C22×#2541 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
84853 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 7 generators on 24 points:
(5,6,8,7)(9,10,14,12)(11,15,16,13)(17,18)(19,23)(20,24)(21,22);(6,7)(11,16)(13,15)(18,22)(20,23);(5,7,8,6)(9,11,14,16)(10,15,12,13)(17,18)(19,20)(21,22)(23,24);(9,12,14,10)(11,15,16,13)(17,19)(18,23)(20,22)(21,24);(9,13,14,15)(10,11,12,16)(17,20,21,23)(18,24,22,19);(3,4);(1,2)(3,4)280 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((5,6,8,7)(9,10,14,12)(11,15,16,13)(17,18)(19,23)(20,24)(21,22), (6,7)(11,16)(13,15)(18,22)(20,23), (5,7,8,6)(9,11,14,16)(10,15,12,13)(17,18)(19,20)(21,22)(23,24), (9,12,14,10)(11,15,16,13)(17,19)(18,23)(20,22)(21,24), (9,13,14,15)(10,11,12,16)(17,20,21,23)(18,24,22,19), (3,4), (1,2)(3,4));