Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C9 · Schur multiplier (H2) = C28 |
| direct product | C2×C9×#539 |
| 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×#121 | C2×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C9 ◁ G | |
1716 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)(5,6)(7,8)(9,10)(13,17)(15,18);(4,8)(5,9)(11,12,14,16)(13,15,17,18);(3,5,7,9)(4,6,8,10)(11,13)(12,15)(14,17)(16,18);(3,6)(4,9)(5,8)(7,10)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2183);