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×#3502 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | C25 |
| commutator, Frattini, cosocle | C22 | C25×C32 |
| prime radical | C23×C32 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ G | |
2892 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)(7,8,10,12)(9,13,14,11)(15,16,20,24)(17,22,18,21)(19,23,27,30)(25,29,26,28);(4,5,6)(15,17,20,18)(16,21,24,22)(19,26,27,25)(23,29,30,28);(7,9,10,14)(8,11,12,13)(15,18,20,17)(16,22,24,21)(19,26,27,25)(23,29,30,28);(15,19,20,27)(16,23,24,30)(17,25,18,26)(21,28,22,29);(15,16)(17,21)(18,22)(19,23)(20,24)(25,28)(26,29)(27,30)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 4562);