Finite Group DatabasebetaFinite Group Databasebeta| order | 486 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6856 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C1 |
| direct product | C2×#6860 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C32 | H(3,𝔽3) |
| commutator, Frattini, cosocle | C33 | C2×C32 |
| prime radical | C2×C33 | C32 |
| LCS | G ▷ C33 ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ C2×C33 ◁ G | |
70 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 162 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2)(4,6,8)(12,13,16,15,18,17,20,14,19);(3,4,7)(5,8,11)(6,10,9)(12,14,16,20,18,19,15,13,17)70 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(486, 67);