Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27×C4 · Schur multiplier (H2) = C227×C4 |
| direct product | C24×#2328 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C27 | C24 |
| commutator | C22 | C27×C4 |
| Frattini, cosocle | C23 | C28 |
| prime radical | C25×#127 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C27 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
2759560 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 1024 |
A minimal faithful permutation representation — 8 generators on 24 points:
(14,16)(17,18)(19,24)(20,23)(21,22);(9,10,12,11)(14,16)(18,22)(19,23);(3,4);(5,6);(13,14)(15,16)(17,19)(18,20)(21,23)(22,24);(7,8);(9,11,12,10)(17,20)(18,23)(19,22)(21,24);(1,2)(3,4)(5,6)(7,8)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((14,16)(17,18)(19,24)(20,23)(21,22), (9,10,12,11)(14,16)(18,22)(19,23), (3,4), (5,6), (13,14)(15,16)(17,19)(18,20)(21,23)(22,24), (7,8), (9,11,12,10)(17,20)(18,23)(19,22)(21,24), (1,2)(3,4)(5,6)(7,8));