Finite Group DatabasebetaFinite Group Databasebeta| order | 405 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AΣL(1,𝔽27) |
| automorphism | Aut(G) = C3×C4×((C3⋊D3)⋊C3)⋊C2 (order 1296) · Inn(G) = H(3,𝔽3) · Out(G) = C22×C3×C4 |
| homology | Abelianization (H1) = C32×C5 · Schur multiplier (H2) = C3 |
| direct product | C5×AΣL(1,𝔽27) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C3×C5 | H(3,𝔽3) |
| commutator, Frattini, cosocle | C32 | C32×C5 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C5 ◁ C32×C5 ◁ G | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 45 |
A minimal faithful permutation representation — 2 generators on 14 points:
(1,2,4)(3,5,8)(6,9,7)(10,11,12,13,14);(2,5,6)85 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(405, 7);