Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#1764 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C8 | D4 |
| commutator | C4 | C2×C42×C5 |
| Frattini, cosocle | C22×C4 | C23×C5 |
| socle | C22×C5 | C4×D4 |
| prime radical | C2×C5×#41 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C8 ◁ C22×C5×C8 ◁ G | |
300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,3,4,5)(6,7)(8,9)(10,11,14,18,15,19,13,16)(12,17,21,24,22,25,20,23);(7,9)(11,16,19,18)(17,23,25,24);(10,12,14,21,15,22,13,20)(11,17,18,24,19,25,16,23)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 5354);