Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#3044 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C5 |
| prime radical | C2×C5×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C5×#117 ◁ G | |
458 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2,3,4,5)(7,9)(10,11,15,19)(12,16,21,30)(13,23,25,33)(14,26,28,18)(17,27,32,24)(20,34,35,41)(22,38,37,31)(29,39,40,36);(6,7,8,9)(10,12)(11,16)(13,20)(14,22)(15,21)(17,29)(18,31)(19,30)(23,34)(24,36)(25,35)(26,38)(27,39)(28,37)(32,40)(33,41);(10,13,15,25)(11,17,19,32)(12,20,21,35)(14,27,28,24)(16,29,30,40)(18,33,26,23)(22,39,37,36)(31,41,38,34);(12,21)(16,30)(20,35)(22,37)(29,40)(31,38)(34,41)(36,39)160 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18721);