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×#3512 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | C25 |
| commutator, Frattini, cosocle | C22 | C25×C32 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ G | |
3096 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)(17,22)(18,23)(19,27)(20,21)(24,28)(25,26)(29,30);(4,5,6)(7,8,10,12)(9,11,13,14)(16,21)(18,22)(24,27)(26,29);(7,9,10,13)(8,11,12,14)(15,17,20,23)(16,22,21,18)(19,25,28,30)(24,29,27,26);(15,18,20,22)(16,23,21,17)(19,26,28,29)(24,30,27,25);(15,19,20,28)(16,24,21,27)(17,25,23,30)(18,26,22,29)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 4572);