Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C28 |
| direct product | C5×#3555 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C25 |
| commutator, Frattini, cosocle | C22 | C25×C5 |
| prime radical | C5×#121 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ G | |
910 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 5 generators on 37 points:
(1,2,3,4,5)(6,7,11,14)(8,9,16,12)(10,18,19,13)(15,20,21,17)(22,23,26,29)(24,27,32,30)(25,33,34,28)(31,36,37,35);(6,8,11,16)(7,12,14,9)(10,15,19,21)(13,20,18,17)(22,23)(24,30)(25,28)(26,29)(27,32)(31,36)(33,34)(35,37);(6,9,11,12)(7,8,14,16)(10,17,19,20)(13,15,18,21);(6,8)(7,12)(9,14)(10,15)(11,16)(13,20)(17,18)(19,21)(22,24)(23,27)(25,31)(26,32)(28,35)(29,30)(33,36)(34,37);(6,10,11,19)(7,13,14,18)(8,15,16,21)(9,17,12,20)(22,25,26,34)(23,28,29,33)(24,31,32,37)(27,35,30,36)190 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21422);