Finite Group DatabasebetaFinite Group Databasebeta| order | 486 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6909 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33 · Schur multiplier (H2) = C33 |
| direct product | C2×#6912 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C3×H(3,𝔽3) |
| commutator, Frattini, cosocle | C32 | C2×C33 |
| prime radical | C2×C33 | C32 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C33 ◁ G | |
124 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)(3,4,7,6,10,17,15,9,21)(5,11,16,14,8,18,13,20,27)(12,23,26,24,22,28,25,19,29);(3,5,12,15,13,25,6,14,24)(4,8,19,9,11,22,10,20,23)(7,16,26,21,27,29,17,18,28);(5,13,14)(8,11,20)(12,24,25)(16,27,18)(19,23,22)(26,28,29)70 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(486, 218);