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×#6858 |
| 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 | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 162 |
A minimal faithful permutation representation — 2 generators on 38 points:
(1,2)(4,6,8)(12,13,17,16,19,27,25,30,36)(14,20,31,22,34,18,28,15,23)(21,35,26,32,24,37,38,33,29);(3,4,7)(5,8,11)(6,10,9)(12,14,13,16,22,19,25,28,30)(15,21,18,20,32,23,34,38,31)(17,26,35,27,37,24,36,29,33)70 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(486, 65);