Finite Group DatabasebetaFinite Group Databasebeta| order | 810 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AΣL(1,𝔽27) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C5 · Schur multiplier (H2) = C3 |
| direct product | C2×C5×#731 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3×C5 | H(3,𝔽3) |
| commutator, Frattini, cosocle | C32 | C2×C32×C5 |
| prime radical | C2×C5×H(3,𝔽3) | C3 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C5 ◁ C2×C32×C5 ◁ G | |
56 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 270 |
A minimal faithful permutation representation — 2 generators on 34 points:
(1,2)(3,4,5,6,7)(8,9,12)(10,16,20)(11,15,21)(13,22,18)(14,25,19)(17,27,30)(23,33,34)(24,32,29)(26,31,28);(8,10,17,11,18,28,19,29,34)(9,13,23,15,24,27,14,16,26)(12,20,30,21,22,31,25,32,33)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(810, 60);