Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C5×#3156 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C5 |
| prime radical | C2×C5×#135 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C5×#117 ◁ G | |
532 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)(12,20,24,34)(13,17,25,31)(14,27,21,18)(16,28,30,23)(22,33,36,41)(26,38,35,32)(29,39,40,37);(10,12,21,28,15,24,14,23)(11,16,27,34,19,30,18,20)(13,22,35,39,25,36,26,37)(17,29,38,41,31,40,32,33);(6,7,8,9)(10,13)(11,17)(12,22)(14,26)(15,25)(16,29)(18,32)(19,31)(20,33)(21,35)(23,37)(24,36)(27,38)(28,39)(30,40)(34,41);(13,25)(17,31)(22,36)(26,35)(29,40)(32,38)(33,41)(37,39)160 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18833);