Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C9 · Schur multiplier (H2) = C25 |
| direct product | C2×C9×#401 |
| subgroup | G/H | |
|---|---|---|
| center | C24×C9 | C23 |
| commutator | C23 | C24×C9 |
| Frattini, cosocle | C23×C3 | C24×C3 |
| socle, prime radical | C24×C3 | C23×C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C9 ◁ G | |
774 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 1152 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2)(27,28,29,30,31,32,33,34,35);(5,9)(7,10)(12,15)(16,17)(19,20,22,24)(21,25,26,23);(3,4,6,8)(5,10,9,7)(11,12,14,15)(13,17,18,16)(19,20)(21,25)(22,24)(23,26);(3,5)(4,7)(6,9)(8,10)(11,13,14,18)(12,16,15,17)(19,21,22,26)(20,23,24,25)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 1132);