Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C32 · Schur multiplier (H2) = C29×C3 |
| direct product | C32×#3500 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | C25 |
| commutator, Frattini, cosocle | C22 | C25×C32 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ G | |
4620 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2,3)(8,10)(11,12)(13,19)(14,22)(15,16)(17,21)(18,23)(20,24)(25,26);(4,5,6)(12,16)(17,19)(18,22)(24,25);(7,8,9,10)(11,13)(12,17)(14,20)(15,21)(16,19)(18,24)(22,25)(23,26);(11,12)(13,17)(14,18)(15,16)(19,21)(20,24)(22,23)(25,26);(11,14,15,23)(12,18,16,22)(13,20,21,26)(17,24,19,25)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 4560);