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×#3567 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C25 |
| commutator, Frattini, cosocle | C22 | C25×C5 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ G | |
950 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)(7,11)(8,14)(13,18)(15,20)(22,23)(24,27)(25,30)(26,31)(28,32)(29,34)(33,35)(36,37);(7,11)(12,17)(13,18)(19,21)(23,24)(25,32)(29,31)(33,37);(6,7)(8,12)(9,13)(10,11)(14,17)(15,19)(16,18)(20,21)(22,24,27,23)(25,30,32,28)(26,31,34,29)(33,36,37,35);(6,8,10,14)(7,12,11,17)(9,15,16,20)(13,19,18,21)(22,25)(23,28)(24,30)(26,33)(27,32)(29,35)(31,36)(34,37);(6,9,10,16)(7,13,11,18)(8,15,14,20)(12,19,17,21)(22,26)(23,29)(24,31)(25,33)(27,34)(28,35)(30,36)(32,37)190 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21434);