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×#6903 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C27 | C32 |
| commutator | C3 | C2×C32×C9 |
| Frattini, cosocle | C9 | C2×C33 |
| socle | C2×C3 | C32×C9 |
| prime radical | C2×H(3,𝔽3) | C9 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C2×C27 ◁ G | |
110 conjugacy classes of subgroups.
| minimal degree | μ(G) = 83 |
| transitive degree | μt(G) = 162 |
A minimal faithful permutation representation — 3 generators on 83 points:
(1,2)(3,4,8,6,10,19,15,24,38,7,11,20,16,25,39,33,47,61,17,26,40,34,48,62,54,68,77)(5,9,18,14,23,37,32,46,60,12,21,35,29,43,57,51,65,74,27,41,55,49,63,72,69,78,81)(13,22,36,31,45,59,53,67,76,28,42,56,50,64,73,70,79,82,30,44,58,52,66,75,71,80,83);(5,12,27)(9,21,41)(13,30,28)(14,29,49)(18,35,55)(22,44,42)(23,43,63)(31,52,50)(32,51,69)(36,58,56)(37,57,72)(45,66,64)(46,65,78)(53,71,70)(59,75,73)(60,74,81)(67,80,79)(76,83,82);(3,5,13)(4,9,22)(6,14,31)(7,12,28)(8,18,36)(10,23,45)(11,21,42)(15,32,53)(16,29,50)(17,27,30)(19,37,59)(20,35,56)(24,46,67)(25,43,64)(26,41,44)(33,51,70)(34,49,52)(38,60,76)(39,57,73)(40,55,58)(47,65,79)(48,63,66)(54,69,71)(61,74,82)(62,72,75)(68,78,80)(77,81,83)198 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(486, 209);