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×#3505 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | C25 |
| commutator, Frattini, cosocle | C22 | C25×C32 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ G | |
3456 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 5 generators on 22 points:
(1,2,3)(7,8)(9,13)(10,11)(12,14)(17,21)(19,22);(4,5,6)(8,11)(12,13)(15,16,18,20)(17,19,21,22);(15,17,18,21)(16,19,20,22);(7,8)(9,12)(10,11)(13,14);(7,9)(8,12)(10,14)(11,13)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 4565);