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×#3509 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | C25 |
| commutator, Frattini, cosocle | C22 | C25×C32 |
| prime radical | C2×C32×#41 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ G | |
2976 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)(9,13)(11,14)(15,16,19,22)(17,20,25,23)(18,26,27,21)(24,29,30,28);(4,5,6)(7,8,10,12)(9,11,13,14)(15,16)(17,23)(18,21)(19,22)(20,25)(24,29)(26,27)(28,30);(7,9,10,13)(8,11,12,14);(15,17)(16,20)(18,24)(19,25)(21,28)(22,23)(26,29)(27,30);(15,18,19,27)(16,21,22,26)(17,24,25,30)(20,28,23,29)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 4569);