Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1913 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C9 · Schur multiplier (H2) = C24 |
| direct product | C9×#1913 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C9 | #386 |
| commutator | C23 | C22×C4×C9 |
| Frattini, cosocle | C24×C3 | C23×C3 |
| socle | C2×C3 | C3×#386 |
| prime radical | C3×#534 | C2×C3 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C9 ◁ C24×C9 ◁ G | |
570 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,3,4,5,6,7,8,9)(10,11)(12,15,19,23)(13,20,21,16)(14,17)(18,22)(24,25);(10,12)(11,15,17,23)(13,18)(14,19)(16,22,20,25)(21,24);(10,13)(11,16)(12,18)(14,21)(15,22)(17,20)(19,24)(23,25)261 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 645);