Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5×C16 · Schur multiplier (H2) = C2 |
| direct product | C5×#1401 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C16 | D4 |
| commutator | C4 | C2×C5×C16 |
| Frattini, cosocle | C2×C16 | C22×C5 |
| socle | C2×C5 | C22⋊C16 |
| prime radical | C5×#41 | C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C5×C16 ◁ C2×C5×C16 ◁ G | |
72 conjugacy classes of subgroups.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 2 generators on 69 points:
(1,2,3,4,5)(6,7,10,16,11,17,29,42,12,18,30,43,32,45,57,65,13,19,31,44,33,46,58,66,9,15,26,39,27,40,54,63)(8,20,22,47,23,48,51,67,21,37,49,41,50,61,68,64,24,38,28,55,52,62,56,69,25,14,34,35,53,36,59,60);(6,8)(7,14)(9,25)(10,28)(11,23)(12,21)(13,24)(15,38)(16,41)(17,36)(18,20)(19,37)(22,31)(26,49)(27,53)(29,56)(30,34)(32,50)(33,52)(35,44)(39,47)(40,62)(42,64)(43,55)(45,48)(46,61)(51,58)(54,68)(57,59)(60,66)(63,67)(65,69)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 534);