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×#3553 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C25 |
| commutator, Frattini, cosocle | C22 | C25×C5 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ G | |
932 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)(8,13)(9,14)(10,18)(11,12)(15,19)(16,17)(20,21)(23,27)(24,30)(25,33)(35,37);(7,12)(9,13)(15,18)(17,20)(22,23,26,27)(24,31,30,28)(25,29,33,34)(32,37,36,35);(6,8,11,14)(7,13,12,9)(10,16,19,21)(15,20,18,17)(22,23)(24,28)(25,29)(26,27)(30,31)(32,35)(33,34)(36,37);(6,9,11,13)(7,14,12,8)(10,17,19,20)(15,21,18,16)(22,24,26,30)(23,28,27,31)(25,32,33,36)(29,35,34,37);(6,10)(7,15)(8,16)(9,17)(11,19)(12,18)(13,20)(14,21)(22,25)(23,29)(24,32)(26,33)(27,34)(28,35)(30,36)(31,37)190 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21420);