Finite Group DatabasebetaFinite Group Databasebeta| order | 405 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | Aut(G) = C4×#4208 (order 648) · Inn(G) = C32 · Out(G) = C3×C4×D3 |
| homology | Abelianization (H1) = C3×C5×C9 · Schur multiplier (H2) = C1 |
| direct product | C5×C27⋊C3 |
| unambiguous semidirect product | C27⋊(C3×C5) |
| unambiguous semidirect product | (C5×C27)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C9 | C32 |
| commutator | C3 | C3×C5×C9 |
| Frattini, cosocle | C9 | C32×C5 |
| socle | C3×C5 | C3×C9 |
| prime radical | C32×C5 | C9 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C5×C9 ◁ G | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 135 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,5,3,7,13,9,16,21,4,8,12,10,17,20,18,24,26,11,6,14,19,15,22,25,23,27)(28,29,30,31,32);(2,6,8)(5,12,14)(7,15,17)(13,20,22)(16,23,24)(21,26,27)165 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(405, 6);