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) = C3 |
| direct product | C2×#6897 |
| 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)(3,4,8)(5,12,22)(6,11,16)(7,9,18)(10,19,14)(13,24,28)(15,21,17)(20,29,26)(23,27,25)(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);(4,9,11)(8,16,18)(10,12,21)(17,22,19)(20,23,24)(27,29,28)(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);(3,5,13,6,14,25,7,15,26)(4,10,20,11,21,24,9,12,23)(8,17,27,16,22,29,18,19,28)(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, 203);