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×#3493 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C9 | C25 |
| commutator | C22 | C25×C9 |
| Frattini, socle, cosocle | C22×C3 | C25×C3 |
| prime radical | C3×#3493 | C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C9 ◁ G | |
1560 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)(20,25)(21,26)(22,30)(23,24)(27,31)(28,29)(32,33);(10,11,13,15)(12,14,16,17)(19,24)(21,25)(27,30)(29,32);(10,12)(11,14)(13,16)(15,17)(18,20,23,26)(19,25,24,21)(22,28,31,33)(27,32,30,29);(18,21,23,25)(19,26,24,20)(22,29,31,32)(27,33,30,28);(18,22)(19,27)(20,28)(21,29)(23,31)(24,30)(25,32)(26,33)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2225);