Finite Group DatabasebetaFinite Group Databasebeta| order | 486 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AΣL(1,𝔽27) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33 · Schur multiplier (H2) = C33 |
| direct product | C2×C3×#733 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C32 | H(3,𝔽3) |
| commutator, Frattini, cosocle | C32 | C2×C33 |
| prime radical | C2×C33 | C32 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ C2×C33 ◁ G | |
132 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 162 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,4,5);(6,7,10,9,13,19,17,12,23)(8,14,18,16,11,20,27,22,30)(15,25,28,26,24,29,32,21,31);(6,8,15,17,27,32,9,16,26)(7,11,21,12,14,24,13,22,25)(10,18,28,23,30,31,19,20,29)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(486, 213);