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×#3556 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C25 |
| commutator, Frattini, cosocle | C22 | C25×C5 |
| prime radical | C23×C5 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ G | |
890 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)(6,7,11,9)(8,14,16,12)(10,17,18,13)(15,19,21,20)(22,23,27,31)(24,29,25,28)(26,34,35,30)(32,36,33,37);(6,8,11,16)(7,12,9,14)(10,15,18,21)(13,19,17,20)(22,23)(24,29)(25,28)(26,30)(27,31)(32,37)(33,36)(34,35);(22,24,27,25)(23,28,31,29)(26,32,35,33)(30,36,34,37);(6,9,11,7)(8,12,16,14)(10,17,18,13)(15,19,21,20)(22,25,27,24)(23,29,31,28)(26,33,35,32)(30,37,34,36);(6,10,11,18)(7,13,9,17)(8,15,16,21)(12,19,14,20)(22,26,27,35)(23,30,31,34)(24,32,25,33)(28,36,29,37)190 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21423);