Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C9 · Schur multiplier (H2) = C25 |
| direct product | C9×#1445 |
| subgroup | G/H | |
|---|---|---|
| center | C24×C9 | C23 |
| commutator | C23 | C22×C4×C9 |
| Frattini, socle, prime radical, cosocle | C24×C3 | C23×C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C9 ◁ G | |
780 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 1152 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,3,4,5,6,7,8,9)(14,15,16,17)(19,21)(23,26)(27,28);(11,13)(15,17)(18,19)(20,21)(22,23,25,26)(24,28,29,27);(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,24,25,29)(23,27,26,28)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 177);