Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C9 · Schur multiplier (H2) = C29 |
| direct product | C4×C9×2−1+4 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C9 | C24 |
| commutator | C2 | C24×C4×C9 |
| Frattini, socle, cosocle | C22×C3 | C25×C3 |
| prime radical | C2×C3×2−1+4 | C2×C3 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C9 ◁ G | |
2238 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 5 generators on 29 points:
(1,2,3,4)(21,22,23,24,25,26,27,28,29);(6,10)(8,16)(11,13)(14,20);(5,6,9,10)(7,13,15,11)(8,12,16,17)(14,19,20,18);(5,7,9,15)(6,11,10,13)(8,14,16,20)(12,18,17,19);(5,8)(6,12)(7,14)(9,16)(10,17)(11,18)(13,19)(15,20)612 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2162);