Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C9 · Schur multiplier (H2) = C25 |
| direct product | C4×C9×#135 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C9 | D4 |
| commutator | C4 | C23×C4×C9 |
| Frattini, cosocle | C2×C3×C4 | C24×C3 |
| socle | C22×C3 | C22×C3×D4 |
| prime radical | C2×C3×#135 | C2×C3 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42×C9 ◁ C2×C42×C9 ◁ G | |
750 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,3,4)(21,22,23,24,25,26,27,28,29);(5,6)(7,14)(8,12)(9,17)(10,13)(11,15)(16,19)(18,20);(5,7)(6,11)(8,15)(9,18)(10,16)(12,19)(13,20)(14,17);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,20,14)504 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 1671);