Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C4 · Schur multiplier (H2) = C3 |
| direct product | C32×#1425 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C4 | D16 |
| commutator | C16 | C2×C32×C4 |
| Frattini, cosocle | C2×C16 | C22×C32 |
| socle | C2×C32 | #366 |
| prime radical | C22×C32 | Q32 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C32×C4 ◁ C2×C32×C4 ◁ C2×C32×C8 ◁ C2×C32×C16 ◁ G | |
150 conjugacy classes of subgroups.
| minimal degree | μ(G) = 70 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 2 generators on 70 points:
(1,2,3)(7,8,11,17,14,19,13,35)(9,20,23,45,26,47,25,57)(10,27,29,58,31,60,30,61)(12,33,32,43,34,62,28,18)(15,36,37,53,39,52,63,22)(16,40,41,50,42,49,59,21)(24,55,54,67,56,70,51,46)(38,48,44,64,65,69,66,68);(4,5,6)(7,9,21,48,12,24,29,36,11,23,40,64,32,54,31,53,14,26,50,69,34,56,30,52,13,25,49,68,28,51,10,22)(8,15,27,46,18,38,59,57,35,63,61,70,62,66,42,47,19,39,60,67,43,65,41,45,17,37,58,55,33,44,16,20)342 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2485);