Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
| direct product | C4×#136361 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #1629 |
| commutator | C4×C8 | C22×C42 |
| Frattini, cosocle | C22×C4×C8 | C24 |
| socle | C23 | C2×#1629 |
| prime radical | C22×#135 | C2×D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ C2×C42×C8 ◁ G | |
| LECS | G ▷ C22×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
4930 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(9,10)(11,14)(12,15)(13,17)(16,24)(18,21)(19,22)(20,23);(1,2,4,3)(11,19)(12,21)(14,22)(15,18)(16,23)(20,24);(1,3,4,2)(10,14,17,23)(11,19)(12,21)(15,16,18,22)(20,24);(5,6,7,8)272 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)(11,14)(12,15)(13,17)(16,24)(18,21)(19,22)(20,23), (1,2,4,3)(11,19)(12,21)(14,22)(15,18)(16,23)(20,24), (1,3,4,2)(10,14,17,23)(11,19)(12,21)(15,16,18,22)(20,24), (5,6,7,8));