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) = C33 |
| direct product | C2×C3×C9⋊C9 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C2×C33 | C32 |
| commutator | C3 | C2×C32×C9 |
| Frattini, cosocle | C32 | C2×C33 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C2×C33 ◁ G | |
180 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 486 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2)(3,4,5);(7,9,12)(10,14,13)(15,16,18,17,19,21,20,22,23);(6,7,10,8,9,13,11,12,14)198 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(486, 192);