Finite Group DatabasebetaFinite Group Databasebeta| order | 528 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C11 · Schur multiplier (H2) = C22 |
| direct product | C3×C11×#41 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4×C11 | C22 |
| commutator, Frattini, cosocle | C2 | C23×C3×C11 |
| socle | C2×C3×C11 | C23 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4×C11 ◁ G | |
80 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 264 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2,3)(4,5,6,7,8,9,10,11,12,13,14)(15,16)(17,20)(18,19)(21,22);(16,19)(20,22);(15,17,18,21)(16,20,19,22)330 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(528, 159);