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×H(5,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C34 |
| commutator, Frattini, cosocle | C3 | C2×C34 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ G | |
586 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)(4,7,9)(5,11,20)(8,16,14)(10,18,15)(12,22,21)(26,28,29);(3,4,8)(5,10,17)(6,7,14)(9,16,13)(11,15,23)(12,19,26)(18,25,20)(21,24,28)(22,27,29);(4,9,7)(8,14,16)(10,18,15)(17,23,25)(19,27,24)(26,28,29);(3,5,12)(4,10,19)(6,11,21)(7,15,24)(8,17,26)(9,18,27)(13,20,22)(14,23,28)(16,25,29)166 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(486, 254);