Finite Group DatabasebetaFinite Group Databasebeta| order | 486 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C9 · Schur multiplier (H2) = C34 |
| direct product | C2×C9×H(3,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C9 | C32 |
| commutator | C3 | C2×C32×C9 |
| Frattini, cosocle | C32 | C2×C33 |
| socle | C2×C32 | C33 |
| prime radical | C2×C3×H(3,𝔽3) | C3 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C9 ◁ G | |
202 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 162 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(12,13,15,14,16,18,17,19,20);(4,6,9)(7,11,10);(3,4,7)(5,6,10)(8,9,11)198 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(486, 194);