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×#3240 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C5 |
| prime radical | C5×#121 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
328 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)(24,32)(25,31)(26,28)(27,30)(29,33)(34,37)(35,36);(6,8,10,16)(7,11,13,14)(9,17,18,15)(12,20,21,19)(23,28)(29,36)(30,31)(32,37);(6,9,10,18)(7,12,13,21)(8,15,16,17)(11,19,14,20)(22,24,26,34)(23,29,28,36)(25,33,27,35)(30,32,31,37);(7,13)(11,14)(12,21)(19,20)(22,25,26,27)(23,30,28,31)(24,33,34,35)(29,32,36,37)130 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18917);