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×#3515 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | C25 |
| commutator, Frattini, cosocle | C22 | C25×C32 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ G | |
3042 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)(15,16,20,21)(17,18,22,25)(19,27,28,23)(24,30,29,26);(4,5,6)(16,21)(18,25)(23,27)(26,30);(7,8)(9,12)(10,11)(13,14)(15,17,20,22)(16,18,21,25)(19,24,28,29)(23,26,27,30);(15,18)(16,22)(17,21)(19,26)(20,25)(23,29)(24,27)(28,30);(7,9,10,13)(8,12,11,14)(15,19)(16,23)(17,24)(18,26)(20,28)(21,27)(22,29)(25,30)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 4575);