Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C2 |
| direct product | C5×#1423 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C5 | D16 |
| commutator | C16 | C2×C4×C5 |
| Frattini, cosocle | C2×C16 | C22×C5 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C4×C5 ◁ C2×C5×C8 ◁ C2×C5×C16 ◁ G | |
56 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 2 generators on 41 points:
(1,2,3,4,5)(6,7,8,9)(11,16)(12,17)(13,21)(14,24)(18,33)(19,36)(20,29)(22,38)(23,32)(25,37)(26,27)(28,34)(30,31)(35,41)(39,40);(10,11,17,27,13,18,30,36,24,34,38,41,37,39,23,29,15,20,32,40,25,35,22,28,14,19,31,33,21,26,12,16)190 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 556);