Finite Group DatabasebetaFinite Group Databasebeta| order | 405 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C33×C5 · Schur multiplier (H2) = C32 |
| direct product | C5×#737 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C9 | C32 |
| commutator, Frattini, cosocle | C3 | C33×C5 |
| socle | C3×C5 | C33 |
| prime radical | C5×H(3,𝔽3) | C3 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C5×C9 ◁ G | |
66 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 135 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5)(3,7,13)(4,8,12)(6,14,11)(9,16,21)(10,17,20)(15,22,19)(18,24,26)(23,27,25)(28,29,30,31,32);(2,6,8)(5,12,14)(7,15,17)(13,20,22)(16,23,24)(21,26,27);(1,3,9,4,10,18,11,19,25)(2,7,16,8,17,24,6,15,23)(5,13,21,12,20,26,14,22,27)165 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(405, 14);