Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C222×C4 |
| direct product | C22×#612904 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C26 | C25 |
| commutator, Frattini, cosocle | C24 | C27 |
| prime radical | C27×D4 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C26 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
8296296 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 7 generators on 24 points:
(9,10)(11,12)(17,18)(19,24)(20,23)(21,22);(9,10)(11,12)(13,14)(15,16)(18,22)(19,23);(6,8)(9,10,11,12)(14,16)(17,19)(18,20)(21,23)(22,24);(13,14)(15,16);(3,4);(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,20)(18,23)(19,22)(21,24);(1,2)(3,4)464 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((9,10)(11,12)(17,18)(19,24)(20,23)(21,22), (9,10)(11,12)(13,14)(15,16)(18,22)(19,23), (6,8)(9,10,11,12)(14,16)(17,19)(18,20)(21,23)(22,24), (13,14)(15,16), (3,4), (5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,20)(18,23)(19,22)(21,24), (1,2)(3,4));