Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C22×C5×Q32 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C5 | D8 |
| commutator, Frattini, cosocle | C8 | C24×C5 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C22×C4×C5 ◁ C22×C5×C8 ◁ G | |
360 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2)(37,38,39,40,41);(3,4);(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22);(5,21,13,29)(6,22,14,30)(7,23,15,31)(8,24,16,32)(9,25,17,33)(10,26,18,34)(11,27,19,35)(12,28,20,36)220 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 19087);