Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2048 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C9 · Schur multiplier (H2) = C23 |
| direct product | C9×#2048 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C9 | #394 |
| commutator | C22×C4 | C23×C9 |
| Frattini, cosocle | C22×C3×C4 | C23×C3 |
| socle | C2×C3 | C3×#394 |
| prime radical | C2×C3×AGL(1,ℤ/8) | C2×C3 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C9 ◁ C22×C4×C9 ◁ G | |
411 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)(11,16)(12,19)(13,14)(17,24)(18,22)(20,21);(10,11)(12,17)(13,22)(14,16)(15,18)(19,23)(20,25)(21,24);(10,12,13,20,15,21,14,19)(11,17,16,23,18,24,22,25)207 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 780);