Finite Group DatabasebetaFinite Group Databasebeta| order | 486 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6909 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33 · Schur multiplier (H2) = C33 |
| direct product | C2×#6909 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C3×H(3,𝔽3) |
| commutator, Frattini, cosocle | C32 | C2×C33 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C33 ◁ G | |
148 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 54 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2)(4,7,14)(5,11,22)(8,19,28)(9,16,17)(10,18,26)(12,24,23)(15,20,25)(21,29,27);(3,4,8)(5,7,15)(6,10,20)(9,21,13)(11,18,27)(12,17,19)(14,25,23)(16,28,22)(24,26,29);(4,9,10)(7,16,18)(8,20,21)(14,17,26)(15,27,28)(19,25,29)70 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(486, 215);