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) = C3×C9 |
| direct product | C2×#6887 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C9 | C32 |
| commutator | C3 | C2×C32×C9 |
| Frattini, cosocle | C32 | C2×C33 |
| socle | C2×C32 | C33 |
| prime radical | C2×C33 | C32 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C9 ◁ G | |
138 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 162 |
A minimal faithful permutation representation — 3 generators on 38 points:
(1,2)(3,4,6,5,7,9,8,10,11)(13,16,22)(17,25,23)(18,24,30)(26,33,31)(27,32,36)(34,38,37);(12,13,17)(14,18,26)(15,16,23)(19,27,34)(20,24,31)(21,22,25)(28,32,37)(29,30,33)(35,36,38);(12,14,19,15,20,28,21,29,35)(13,18,27,16,24,32,22,30,36)(17,26,34,23,31,37,25,33,38)198 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(486, 193);