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×#731 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C3×C5 | H(3,𝔽3) |
| commutator, Frattini, cosocle | C32 | C32×C5 |
| prime radical | C5×H(3,𝔽3) | C3 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C5 ◁ C32×C5 ◁ G | |
28 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 135 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,5)(3,9,13)(4,8,14)(6,15,11)(7,18,12)(10,20,23)(16,26,27)(17,25,22)(19,24,21)(28,29,30,31,32);(1,3,10,4,11,21,12,22,27)(2,6,16,8,17,20,7,9,19)(5,13,23,14,15,24,18,25,26)85 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(405, 8);