Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C9 · Schur multiplier (H2) = C23 |
| direct product | C9×#1839 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C9 | C23 |
| commutator | C22 | C22×C8×C9 |
| Frattini, cosocle | C22×C3×C4 | C23×C3 |
| socle, prime radical | C23×C3 | C22×C3×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C9 ◁ G | |
360 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 1152 |
A minimal faithful permutation representation — 3 generators on 33 points:
(1,2,3,4,5,6,7,8,9)(10,11)(12,17)(13,15)(14,16)(18,19)(20,22)(21,23)(24,25)(26,27,28,30,29,31,32,33);(10,12,14,13)(11,15,16,17)(18,20,21,24)(19,22,23,25);(10,13,14,12)(11,15,16,17)(19,23)(22,25)504 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 571);