Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C9 · Schur multiplier (H2) = C28 |
| direct product | C9×#3491 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C9 | C25 |
| commutator | C22 | C25×C9 |
| Frattini, socle, cosocle | C22×C3 | C25×C3 |
| prime radical | C3×#534 | C2×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C9 ◁ G | |
1578 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 5 generators on 33 points:
(1,2,3,4,5,6,7,8,9)(12,16)(14,17)(18,19)(20,24)(21,28)(22,23)(25,29)(26,32)(27,30)(31,33);(10,11,13,15)(12,14,16,17)(19,23)(24,30)(25,28)(31,32);(10,12)(11,14)(13,16)(15,17)(18,20,22,27)(19,24,23,30)(21,26,29,33)(25,31,28,32);(18,19)(20,24)(21,25)(22,23)(26,31)(27,30)(28,29)(32,33);(18,21)(19,25)(20,26)(22,29)(23,28)(24,31)(27,33)(30,32)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2223);