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×#3495 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C9 | C25 |
| commutator | C22 | C25×C9 |
| Frattini, socle, cosocle | C22×C3 | C25×C3 |
| prime radical | C3×#525 | C2×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C9 ◁ G | |
1533 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)(18,19,23,26)(20,21,28,24)(22,30,31,25)(27,32,33,29);(10,11,13,15)(12,14,16,17)(18,20,23,28)(19,24,26,21)(22,27,31,33)(25,32,30,29);(10,12)(11,14)(13,16)(15,17)(18,21)(19,20)(22,29)(23,24)(25,27)(26,28)(30,33)(31,32);(18,20)(19,21)(22,27)(23,28)(24,26)(25,29)(30,32)(31,33);(18,22,23,31)(19,25,26,30)(20,27,28,33)(21,29,24,32)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2227);