Finite Group DatabasebetaFinite Group Databasebeta| order | 486 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of C32⋊H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33 · Schur multiplier (H2) = C9 |
| direct product | C2×#6898 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C32 | C33 |
| commutator, Frattini, cosocle | C32 | C2×C33 |
| prime radical | C2×C33 | C32 |
| LCS | G ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ G | |
100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 162 |
A minimal faithful permutation representation — 3 generators on 56 points:
(1,2)(3,4,7)(5,9,15)(6,10,14)(8,16,13)(11,18,23)(12,19,22)(17,24,21)(20,26,28)(25,29,27)(30,31,34)(32,35,41)(33,37,42)(36,46,40)(38,43,49)(39,45,50)(44,53,48)(47,52,55)(51,56,54);(4,8,10)(7,14,16)(9,17,19)(15,22,24)(18,25,26)(23,28,29)(30,32,38,33,39,47,40,48,54)(31,35,43,37,45,52,36,44,51)(34,41,49,42,50,55,46,53,56);(3,5,11,13,21,27,6,12,20)(4,9,18,8,17,25,10,19,26)(7,15,23,16,24,29,14,22,28)(31,36,37)(34,42,46)(35,44,45)(41,50,53)(43,51,52)(49,55,56)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(486, 204);