Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#3177 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C5 |
| prime radical | C23×C5 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C5×#117 ◁ G | |
338 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2,3,4,5)(6,7)(8,11)(9,10)(12,13)(14,15,19,23)(16,20,25,33)(17,27,30,42)(18,31,28,22)(21,35,36,29)(24,37,39,45)(26,41,38,34)(32,43,44,40);(7,10)(11,13)(14,16,19,25)(15,20,23,33)(17,24,30,39)(18,26,28,38)(21,32,36,44)(22,34,31,41)(27,37,42,45)(29,40,35,43);(14,17,28,35,19,30,18,29)(15,21,31,42,23,36,22,27)(16,24,38,43,25,39,26,40)(20,32,41,45,33,44,34,37);(6,8,9,12)(7,11,10,13)(16,25)(20,33)(24,39)(26,38)(32,44)(34,41)(37,45)(40,43)160 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18854);