Finite Group DatabasebetaFinite Group Databasebeta| order | 405 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AΣL(1,𝔽27) |
| automorphism | Aut(G) = C4×#92285 (order 1944) · Inn(G) = H(3,𝔽3) · Out(G) = C2×C32×C4 |
| homology | Abelianization (H1) = C32×C5 · Schur multiplier (H2) = C3 |
| direct product | C5×#733 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C3×C5 | H(3,𝔽3) |
| commutator, Frattini, cosocle | C32 | C32×C5 |
| prime radical | C32×C5 | C32 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C5 ◁ C32×C5 ◁ G | |
26 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 135 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,5,4,8,14,12,7,18)(3,9,13,11,6,15,22,17,25)(10,20,23,21,19,24,27,16,26)(28,29,30,31,32);(1,3,10,12,22,27,4,11,21)(2,6,16,7,9,19,8,17,20)(5,13,23,18,25,26,14,15,24)85 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(405, 10);