Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C25×C3 |
| direct product | C2×C32×#464 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C32 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C32 |
| prime radical | C22×C32×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C32 ◁ C2×C32×C42 ◁ G | |
1500 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(23,24,25);(4,6)(7,8)(9,15)(10,12)(11,19)(13,17)(14,18)(16,22)(20,21)(26,27,28);(3,4,5,6)(8,12)(13,21)(14,19)(15,22);(7,9,10,16)(8,13,12,21)(11,20,18,17)(14,22,19,15)342 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 4094);