Finite Group DatabasebetaFinite Group Databasebeta| order | 486 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AΣL(1,𝔽27) |
| automorphism | Aut(G) = C9×((C3⋊D3)⋊C3)⋊C2 (order 972) · Inn(G) = H(3,𝔽3) · Out(G) = C22×C9 |
| homology | Abelianization (H1) = C2×C3×C9 · Schur multiplier (H2) = C3 |
| direct product | C2×#6873 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C9 | H(3,𝔽3) |
| commutator | C32 | C2×C3×C9 |
| Frattini, cosocle | C3×C9 | C2×C32 |
| socle | C2×C3 | C32⋊C9 |
| prime radical | C2×H(3,𝔽3) | C9 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C2×C9 ◁ C2×C3×C9 ◁ G | |
52 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)(3,4,8,6,11,22,17,30,46,19,32,48,42,59,70,65,78,82,7,12,23,18,31,47,41,58,69)(5,13,20,16,33,43,40,60,66,34,10,28,54,29,56,75,57,77,14,25,21,36,51,45,61,73,68)(9,24,49,26,50,71,52,72,15,37,55,39,62,76,64,79,83,38,27,44,63,53,67,80,74,81,35);(4,9,25)(5,14,34)(8,20,24)(10,12,27)(11,26,51)(13,32,37)(15,38,35)(16,36,54)(21,44,23)(22,43,50)(28,55,48)(29,31,53)(30,52,73)(33,59,62)(39,63,49)(40,61,75)(45,67,47)(46,66,72)(56,76,70)(57,58,74)(60,78,79)(64,80,71)(68,81,69)(77,83,82)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(486, 80);