Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C9 · Schur multiplier (H2) = C23 |
| direct product | C2×C9×D32 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C9 | D16 |
| commutator | C16 | C23×C9 |
| Frattini, cosocle | C3×C16 | C23×C3 |
| socle | C22×C3 | C3×D16 |
| prime radical | C2×C3×D32 | C3 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C9 ◁ C2×C4×C9 ◁ C2×C8×C9 ◁ C2×C9×C16 ◁ G | |
180 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2)(35,36,37,38,39,40,41,42,43);(3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34);(4,34)(5,33)(6,32)(7,31)(8,30)(9,29)(10,28)(11,27)(12,26)(13,25)(14,24)(15,23)(16,22)(17,21)(18,20)342 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 991);