Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27×C4 · Schur multiplier (H2) = C229 |
| direct product | C24×#2308 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C27 | C24 |
| commutator | C22 | C27×C4 |
| Frattini, cosocle | C23 | C28 |
| prime radical | C27×D4 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C27 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
9854440 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 8 generators on 24 points:
(3,4);(5,6);(7,8);(10,12)(13,14,15,16)(17,18,20,21)(19,22,24,23);(1,2)(3,4)(5,6)(7,8);(10,12)(13,14,15,16)(18,21)(22,23);(9,10)(11,12)(13,14)(15,16)(17,18)(19,23)(20,21)(22,24);(17,19)(18,22)(20,24)(21,23)704 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((3,4), (5,6), (7,8), (10,12)(13,14,15,16)(17,18,20,21)(19,22,24,23), (1,2)(3,4)(5,6)(7,8), (10,12)(13,14,15,16)(18,21)(22,23), (9,10)(11,12)(13,14)(15,16)(17,18)(19,23)(20,21)(22,24), (17,19)(18,22)(20,24)(21,23));