Finite Group DatabasebetaFinite Group Databasebeta| order | 486 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C9 · Schur multiplier (H2) = C32 |
| direct product | C2×C3×C27⋊C3 |
| unambiguous semidirect product | (C2×C27)⋊C32 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C9 | C32 |
| commutator | C3 | C2×C32×C9 |
| Frattini, cosocle | C9 | C2×C33 |
| socle | C2×C32 | C3×C9 |
| prime radical | C2×C33 | C9 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C9 ◁ G | |
120 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);(6,7,10,8,12,18,14,21,26,9,13,17,15,22,25,23,29,31,16,11,19,24,20,27,30,28,32);(7,11,13)(10,17,19)(12,20,22)(18,25,27)(21,28,29)(26,31,32)198 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(486, 208);