Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C9 · Schur multiplier (H2) = C2×C4 |
| direct product | C9×#2247 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C9 | C2×D8 |
| commutator | C2×C8 | C23×C9 |
| Frattini, cosocle | C2×C3×C8 | C23×C3 |
| socle, prime radical | C22×C3 | C2×C3×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C9 ◁ C42×C9 ◁ C4×C8×C9 ◁ G | |
264 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 1152 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5,6,7,8,9)(11,13)(14,15,19,23)(16,24,28,40)(17,29,31,44)(18,32,30,22)(20,33,35,26)(21,36,37,25)(27,43,42,39)(34,41,38,45);(14,16,25,41,18,27,31,33,19,28,36,45,30,42,17,26)(15,20,29,39,22,34,37,40,23,35,44,43,32,38,21,24);(10,11,12,13)342 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 979);