Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C32 · Schur multiplier (H2) = C28×C3 |
| direct product | C32×#3518 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | C25 |
| commutator, Frattini, cosocle | C22 | C25×C32 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ G | |
3030 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 5 generators on 30 points:
(1,2,3)(8,11)(12,14)(15,16,21,19)(17,24,18,22)(20,27,28,23)(25,29,26,30);(4,5,6)(15,17,21,18)(16,22,19,24)(20,25,28,26)(23,29,27,30);(7,8)(9,12)(10,11)(13,14)(15,18,21,17)(16,22,19,24)(20,26,28,25)(23,29,27,30);(15,19,21,16)(17,22,18,24)(20,27,28,23)(25,29,26,30);(7,9,10,13)(8,12,11,14)(15,20)(16,23)(17,25)(18,26)(19,27)(21,28)(22,29)(24,30)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 4578);