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×#731 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C32 | H(3,𝔽3) |
| commutator, Frattini, cosocle | C32 | C2×C33 |
| prime radical | C2×C3×H(3,𝔽3) | C3 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ C2×C33 ◁ G | |
144 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);(7,10,16)(8,13,23)(11,20,30)(12,19,27)(14,25,24)(17,21,26)(18,22,29)(28,31,32);(6,7,11,9,12,21,15,22,31)(8,10,17,13,19,28,23,29,30)(14,18,20,24,16,26,25,27,32)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(486, 211);