Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C9 · Schur multiplier (H2) = C24 |
| direct product | C9×#3127 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C9 | C22×D4 |
| commutator | C2×C4 | C24×C9 |
| Frattini, cosocle | C2×C3×C4 | C24×C3 |
| socle | C22×C3 | C22×C3×D4 |
| prime radical | C23×C3 | C2×C3×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C9 ◁ C9×#117 ◁ G | |
537 conjugacy classes of subgroups.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 4 generators on 49 points:
(1,2,3,4,5,6,7,8,9)(10,11)(12,15)(13,14)(16,17)(18,19,22,26)(20,28,30,24)(21,32,33,41)(23,36,35,27)(25,37,40,34)(29,44,45,49)(31,39,47,43)(38,46,42,48);(11,14)(15,17)(18,20)(19,24)(21,29)(22,30)(23,31)(25,38)(26,28)(27,39)(32,49)(33,45)(34,46)(35,47)(36,43)(37,48)(40,42)(41,44);(18,21,22,33)(19,25,26,40)(20,29,30,45)(23,37,35,34)(24,38,28,42)(27,41,36,32)(31,48,47,46)(39,44,43,49);(10,12,13,16)(11,15,14,17)(20,30)(24,28)(29,45)(31,47)(38,42)(39,43)(44,49)(46,48)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 1859);