Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C9 · Schur multiplier (H2) = C25 |
| direct product | C9×#2929 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C9 | C23 |
| commutator | C22 | C23×C4×C9 |
| Frattini, cosocle | C2×C3×C4 | C24×C3 |
| socle | C22×C3 | C23×C3×C4 |
| prime radical | C3×#525 | C2×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C9 ◁ G | |
780 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,3,4,5,6,7,8,9)(10,11)(12,13)(14,15,17,21,18,20,24,27)(16,19,22,26,23,25,28,29);(11,13);(14,16,17,22,18,23,24,28)(15,19,21,26,20,25,27,29);(15,20)(19,25)(21,27)(26,29)504 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 1661);