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×#6908 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C9 | H(3,𝔽3) |
| commutator, Frattini, cosocle | C32 | C2×C33 |
| socle | C2×C3 | C3×H(3,𝔽3) |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C2×C9 ◁ C2×C3×C9 ◁ G | |
134 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 54 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2)(3,4,7)(5,9,14)(6,8,15)(10,19,13)(11,17,22)(12,16,23)(18,26,21)(20,24,28)(25,29,27);(4,8,10)(9,16,18)(17,24,25);(3,5,11,6,12,20,13,21,27)(4,9,17,8,16,24,10,18,25)(7,14,22,15,23,28,19,26,29)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(486, 214);