Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C28 · Schur multiplier (H2) = C226 |
| direct product | C22×#7849930 |
| 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 10116 conjugacy classes of subgroups; only the 445264 normal subgroups are shown.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 8 generators on 24 points:
(6,8)(10,14)(12,15);(5,6)(7,8)(9,10,13,14)(11,15,16,12)(17,18,20,22)(19,23,24,21);(10,14)(11,16)(19,24)(21,23);(10,14)(11,16)(17,19)(18,21)(20,24)(22,23);(9,11,13,16)(10,12,14,15)(17,18)(19,23)(20,22)(21,24);(9,12)(10,11)(13,15)(14,16);(3,4);(1,2)(3,4)404 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((6,8)(10,14)(12,15), (5,6)(7,8)(9,10,13,14)(11,15,16,12)(17,18,20,22)(19,23,24,21), (10,14)(11,16)(19,24)(21,23), (10,14)(11,16)(17,19)(18,21)(20,24)(22,23), (9,11,13,16)(10,12,14,15)(17,18)(19,23)(20,22)(21,24), (9,12)(10,11)(13,15)(14,16), (3,4), (1,2)(3,4));