Finite Group DatabasebetaFinite Group Databasebeta| order | 486 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | Aut(G) = #92319 (order 486) · Inn(G) = C32 · Out(G) = D3×C9 |
| homology | Abelianization (H1) = C2×C3×C27 · Schur multiplier (H2) = C1 |
| direct product | C2×C81⋊C3 |
| unambiguous semidirect product | (C2×C81)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C27 | C32 |
| commutator | C3 | C2×C3×C27 |
| Frattini, cosocle | C27 | C2×C32 |
| socle | C2×C3 | C3×C27 |
| prime radical | C2×C32 | C27 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C2×C27 ◁ G | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 83 |
| transitive degree | μt(G) = 162 |
A minimal faithful permutation representation — 2 generators on 83 points:
(1,2)(3,4,8,5,10,20,13,25,39,6,11,21,14,26,40,30,47,61,16,28,41,32,49,62,52,68,76,7,12,19,15,27,37,31,48,58,17,29,38,33,50,59,53,69,74,35,51,60,55,70,75,71,80,82,18,9,22,34,23,42,54,44,63,36,24,43,56,45,64,72,66,77,57,46,65,73,67,78,81,79,83);(4,9,12)(8,19,22)(10,23,27)(11,24,29)(20,37,42)(21,38,43)(25,44,48)(26,45,50)(28,46,51)(39,58,63)(40,59,64)(41,60,65)(47,66,69)(49,67,70)(61,74,77)(62,75,78)(68,79,80)(76,82,83)198 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(486, 84);