Finite Group DatabasebetaFinite Group Databasebeta| order | 486 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #6881 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C3 |
| direct product | C2×#6882 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #732 |
| commutator, Frattini, cosocle | C3×C9 | C2×C32 |
| prime radical | C2×C32 | H(3,𝔽3) |
| LCS | G ▷ C3×C9 ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C32 ◁ C2×C3×C9 ◁ G | |
38 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)(3,4,7,6,11,17,15,10,23)(5,12,16,13,9,21,24,22,29)(8,18,25,20,27,26,19,28,14);(4,8,9,11,20,22,10,19,12)(5,13,24)(7,16,27,23,29,18,17,21,28)(14,26,25)38 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(486, 89);