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) = C24 |
| direct product | C2×C5×#341 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | D4 |
| commutator | C4 | C2×C42×C5 |
| Frattini, cosocle | C22×C4 | C23×C5 |
| socle | C23×C5 | C4⋊C4 |
| prime radical | C24×C5 | Q8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ C23×C4×C5 ◁ G | |
392 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2)(23,24,25,26,27);(3,4)(5,6)(7,8,10,15,12,16,11,14)(9,17,18,13,20,21,19,22);(4,6)(7,9,11,19,12,20,10,18)(8,13,15,21,16,22,14,17)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 5327);