Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C28 · Schur multiplier (H2) = C226 |
| direct product | C22×#7768137 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C26 |
| commutator, Frattini, cosocle | C23 | C28 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
At least 9633 conjugacy classes of subgroups; only the 445480 normal subgroups are shown.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 8 generators on 24 points:
(9,10,13,14)(11,15,16,12)(17,18,20,22)(19,23,24,21);(10,14)(11,16)(17,19)(18,21)(20,24)(22,23);(9,11,13,16)(10,15,14,12)(17,18,20,22)(19,21,24,23);(3,4);(9,12)(10,11)(13,15)(14,16)(17,18)(19,21)(20,22)(23,24);(1,2)(3,4);(6,8)(9,12)(10,11)(13,15)(14,16);(5,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,21)(20,24)(22,23)392 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((9,10,13,14)(11,15,16,12)(17,18,20,22)(19,23,24,21), (10,14)(11,16)(17,19)(18,21)(20,24)(22,23), (9,11,13,16)(10,15,14,12)(17,18,20,22)(19,21,24,23), (3,4), (9,12)(10,11)(13,15)(14,16)(17,18)(19,21)(20,22)(23,24), (1,2)(3,4), (6,8)(9,12)(10,11)(13,15)(14,16), (5,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,21)(20,24)(22,23));