Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C220×C4 |
| direct product | C22×#1083597 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C26 | C25 |
| commutator, Frattini, cosocle | C24 | C27 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C26 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
At least 7818 conjugacy classes of subgroups; only the 42384 normal subgroups are shown.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 7 generators on 24 points:
(13,14,15,16)(18,21)(19,23);(3,4);(6,7)(10,12)(13,14)(15,16)(19,23)(22,24);(1,2)(3,4);(6,7)(18,21)(22,24);(5,6,8,7)(9,10)(11,12)(17,18,20,21)(19,22,23,24);(5,7,8,6)(17,19)(18,22)(20,23)(21,24)344 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((13,14,15,16)(18,21)(19,23), (3,4), (6,7)(10,12)(13,14)(15,16)(19,23)(22,24), (1,2)(3,4), (6,7)(18,21)(22,24), (5,6,8,7)(9,10)(11,12)(17,18,20,21)(19,22,23,24), (5,7,8,6)(17,19)(18,22)(20,23)(21,24));