Finite Group DatabasebetaFinite Group Databasebeta| order | 810 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33×C5 · Schur multiplier (H2) = C32 |
| direct product | C2×C5×#737 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C9 | C32 |
| commutator, Frattini, cosocle | C3 | C2×C33×C5 |
| socle | C2×C3×C5 | C33 |
| prime radical | C2×C5×H(3,𝔽3) | C3 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C9 ◁ G | |
132 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 270 |
A minimal faithful permutation representation — 3 generators on 34 points:
(1,2)(3,4,5,6,7)(8,9,12)(10,14,20)(11,15,19)(13,21,18)(16,23,28)(17,24,27)(22,29,26)(25,31,33)(30,34,32);(9,13,15)(12,19,21)(14,22,24)(20,27,29)(23,30,31)(28,33,34);(8,10,16,11,17,25,18,26,32)(9,14,23,15,24,31,13,22,30)(12,20,28,19,27,33,21,29,34)330 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(810, 100);