Finite Group DatabasebetaFinite Group Databasebeta| order | 486 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6875 |
| automorphism | Aut(G) = #6895⋊C2 (order 486) · Inn(G) = C9⋊C9 · Out(G) = D3 |
| homology | Abelianization (H1) = C2×C3×C9 · Schur multiplier (H2) = C1 |
| direct product | C2×#6875 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C9⋊C9 |
| commutator | C9 | C2×C3×C9 |
| Frattini, cosocle | C3×C9 | C2×C32 |
| prime radical | C2×C32 | 3−1+2 |
| LCS | G ▷ C9 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C3×C9 ◁ G | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 54 |
A minimal faithful permutation representation — 2 generators on 29 points:
(1,2)(4,7,13,9,16,25,20,28,15)(5,10,21)(8,17,14,18,29,26,19,24,27)(11,23,22);(3,4,8,5,7,14,11,15,17,6,9,19,10,16,27,22,13,24,12,20,18,21,28,26,23,25,29)70 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(486, 82);