Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32×C4 · Schur multiplier (H2) = C29×C3 |
| direct product | C32×C4×H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32×C4 | C24 |
| commutator | C2 | C24×C32×C4 |
| Frattini, cosocle | C22 | C25×C32 |
| socle | C22×C32 | C25 |
| prime radical | C2×C32×H(5,𝔽2) | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32×C4 ◁ G | |
4980 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 5 generators on 18 points:
(1,2,3)(7,8,9,10);(4,5,6)(11,12)(13,17)(14,15)(16,18);(12,15)(16,17);(11,12)(13,16)(14,15)(17,18);(11,13)(12,16)(14,18)(15,17)612 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 4489);