Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3192 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#3237 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C5 |
| prime radical | C23×C5 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
322 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7)(8,14)(9,12)(10,13)(11,16)(15,20)(17,19)(18,21)(22,23,27,30)(24,31,32,37)(25,29,26,34)(28,35,36,33);(6,8,10,16)(7,11,13,14)(9,17,18,15)(12,20,21,19)(24,32)(28,36)(31,37)(33,35);(6,9,10,18)(7,12,13,21)(8,15,16,17)(11,19,14,20)(22,24)(23,28)(25,33)(26,35)(27,32)(29,37)(30,36)(31,34);(7,13)(11,14)(12,21)(19,20)(22,25,27,26)(23,29,30,34)(24,33,32,35)(28,37,36,31)130 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18914);