Finite Group DatabasebetaFinite Group Databasebeta| order | 486 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C34 · Schur multiplier (H2) = C35 |
| direct product | C2×3−1+4 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C34 |
| commutator, Frattini, cosocle | C3 | C2×C34 |
| prime radical | C2×C3×H(3,𝔽3) | C3 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ G | |
460 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 54 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2)(3,4,7,6,10,14,13,8,16)(5,9,15,11,19,23,20,17,24)(12,18,25,21,27,28,22,26,29);(4,8,10)(5,11,20)(7,14,16)(12,22,21)(15,24,23)(18,27,26);(3,5,12)(4,9,18)(6,11,21)(7,15,25)(8,17,26)(10,19,27)(13,20,22)(14,23,28)(16,24,29);(4,8,10)(7,14,16)(9,17,19)(15,23,24)(18,26,27)(25,28,29)166 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(486, 255);