Finite Group DatabasebetaFinite Group Databasebeta| order | 486 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C27 · Schur multiplier (H2) = C32 |
| direct product | C2×C32⋊C27 |
| unambiguous semidirect product | (C2×C32)⋊C27 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C9 | C32 |
| commutator | C3 | C2×C3×C27 |
| Frattini, cosocle | C3×C9 | C2×C32 |
| socle | C2×C32 | C3×C9 |
| prime radical | C2×C33 | C9 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C9 ◁ G | |
70 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 162 |
A minimal faithful permutation representation — 2 generators on 38 points:
(1,2)(4,6,9)(7,11,10)(12,13,16,14,17,22,19,24,29,15,18,23,20,25,30,27,32,35,21,26,31,28,33,36,34,37,38);(3,4,7)(5,6,10)(8,9,11)198 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(486, 72);