Finite Group DatabasebetaFinite Group Databasebeta| order | 528 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) order 3840 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C23 |
| abelian | C22×C3×C4×C11 |
| multiplicative | (ℤ/805)ˣ, (ℤ/897)ˣ, (ℤ/1005)ˣ, (ℤ/1035)ˣ, (ℤ/1072)ˣ, (ℤ/1196)ˣ, (ℤ/1340)ˣ, (ℤ/1610)ˣ, (ℤ/1794)ˣ, (ℤ/2010)ˣ, (ℤ/2070)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C2 | C23×C3×C11 |
| socle, prime radical | C23×C3×C11 | C2 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 528 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2,3,4)(9,10,11)(12,13,14,15,16,17,18,19,20,21,22);(5,6);(7,8)528 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(528, 156);