Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C25×C3 |
| direct product | C32×#3414 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C4 | C2×D8 |
| commutator, Frattini, cosocle | C8 | C24×C32 |
| socle | C2×C32 | C22×D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C32×C4 ◁ C22×C32×C4 ◁ C22×C32×C8 ◁ G | |
1092 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2,3)(7,8)(9,18)(10,16)(11,23)(12,27)(13,15)(14,20)(17,25)(19,35)(21,34)(22,32)(24,30)(26,37)(28,33)(29,36)(31,38);(4,5,6)(7,9)(8,14)(10,19)(11,24)(12,28)(13,21)(15,29)(16,30)(17,33)(18,23)(20,25)(22,36)(26,38)(27,31)(32,35)(34,37);(8,15)(14,29)(16,27)(17,32)(18,34)(23,37)(30,31)(33,35);(7,10,13,12)(8,16,15,27)(9,19,21,28)(11,22,26,25)(14,30,29,31)(17,23,32,37)(18,35,34,33)(20,24,36,38)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 4474);