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×#3510 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | C25 |
| commutator, Frattini, cosocle | C22 | C25×C32 |
| prime radical | C32×#535 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ G | |
3120 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)(9,13)(11,14)(15,16)(17,21)(18,25)(19,20)(22,26)(23,29)(24,27)(28,30);(4,5,6)(7,8,10,12)(9,11,13,14)(16,20)(21,27)(22,25)(28,29);(7,9,10,13)(8,11,12,14)(15,17,19,24)(16,21,20,27)(18,23,26,30)(22,28,25,29);(15,16)(17,21)(18,22)(19,20)(23,28)(24,27)(25,26)(29,30);(15,18,19,26)(16,22,20,25)(17,23,24,30)(21,28,27,29)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 4570);