Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of D64 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C1 |
| direct product | C5×SD128 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | D32 |
| commutator, Frattini, cosocle | C32 | C22×C5 |
| prime radical | C5×D32 | C2 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C4×C5 ◁ C5×C8 ◁ C5×C16 ◁ C5×C32 ◁ G | |
38 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)(6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69);(7,37)(8,68)(9,35)(10,66)(11,33)(12,64)(13,31)(14,62)(15,29)(16,60)(17,27)(18,58)(19,25)(20,56)(21,23)(22,54)(24,52)(26,50)(28,48)(30,46)(32,44)(34,42)(36,40)(39,69)(41,67)(43,65)(45,63)(47,61)(49,59)(51,57)(53,55)175 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 563);