Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#2248 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23×C5 |
| prime radical | C5×#493 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C42×C5 ◁ C4×C5×C8 ◁ G | |
196 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,3,4,5)(7,9)(11,15)(12,19)(13,20)(16,21)(17,25)(18,24)(22,23);(10,11,16,25,13,17,12,15,14,18,19,23,20,22,21,24);(6,7,8,9)190 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 5838);