Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C9 · Schur multiplier (H2) = C25 |
| direct product | C9×#2327 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C9 | C24 |
| commutator | C22 | C23×C4×C9 |
| Frattini, socle, prime radical, cosocle | C23×C3 | C24×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C9 ◁ G | |
774 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 1152 |
A minimal faithful permutation representation — 4 generators on 33 points:
(1,2,3,4,5,6,7,8,9)(11,13)(14,15,17,16)(19,23)(20,26)(21,29)(31,33);(14,16,17,15)(18,19)(20,27)(21,25)(22,23)(24,26)(28,33)(29,30)(31,32);(10,11,12,13)(18,20)(19,24)(21,28)(22,26)(23,27)(25,31)(29,32)(30,33);(18,21,22,29)(19,25,23,30)(20,28,26,32)(24,31,27,33)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 1059);