Finite Group DatabasebetaFinite Group Databasebeta| order | 810 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C5×C9 · Schur multiplier (H2) = C32 |
| direct product | C2×C5×C32⋊C9 |
| unambiguous semidirect product | (C2×C32)⋊(C5×C9) |
| unambiguous semidirect product | (C2×C32×C5)⋊C9 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C32×C5 | C32 |
| commutator | C3 | C2×C3×C5×C9 |
| Frattini, cosocle | C32 | C2×C32×C5 |
| prime radical | C2×C33×C5 | C3 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C2×C32×C5 ◁ G | |
92 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 270 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,2)(3,4,5,6,7)(8,9,11)(10,13,15)(12,16,14)(17,18,20,19,21,23,22,24,25);(9,12,13)(11,15,16)330 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(810, 55);