Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C9 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×C9×#555 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C9 | C24 |
| commutator | C22 | C25×C9 |
| Frattini, cosocle | C22×C3 | C25×C3 |
| socle | C23×C3 | C24×C3 |
| prime radical | C2×C3×#555 | C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C9 ◁ G | |
1860 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2)(19,20,21,22,23,24,25,26,27);(3,4)(5,9)(6,10)(7,8)(12,13)(16,17);(4,8)(6,9)(11,12,15,13)(14,17,18,16);(3,5,7,10)(4,9,8,6)(11,13,15,12)(14,17,18,16);(3,6,7,9)(4,10,8,5)(11,14,15,18)(12,16,13,17)450 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2205);