Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C32 · Schur multiplier (H2) = C28×C3 |
| direct product | C32×#3529 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32×C4 | C24 |
| commutator, Frattini, cosocle | C22 | C25×C32 |
| socle | C22×C32 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C32×C4 ◁ G | |
3174 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 5 generators on 30 points:
(1,2,3)(7,8)(9,11)(10,12)(13,14)(15,16)(17,23)(18,22)(19,21)(20,24)(25,29)(26,28)(27,30);(4,5,6)(7,9)(8,11)(10,13)(12,14)(15,17)(16,20)(18,25)(19,24)(21,23)(22,27)(26,30)(28,29);(8,12)(9,13)(17,24)(20,23)(25,30)(27,29);(8,12)(11,14)(16,21)(17,24)(22,28)(25,30);(15,18,19,26)(16,22,21,28)(17,25,24,30)(20,27,23,29)342 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 4589);