Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C9 · Schur multiplier (H2) = C26 |
| direct product | C2×C9×#397 |
| subgroup | G/H | |
|---|---|---|
| center | C24×C9 | C23 |
| commutator | C23 | C24×C9 |
| Frattini, cosocle | C23×C3 | C24×C3 |
| socle | C24×C3 | C23×C3 |
| prime radical | C25×C3 | C22×C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C9 ◁ G | |
1230 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2)(19,20,21,22,23,24,25,26,27);(7,8,9,10)(12,15)(16,17);(4,6)(11,12)(13,17)(14,15)(16,18);(3,4)(5,6)(7,8)(9,10)(11,13,14,18)(12,16,15,17)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 1122);