Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×Q8×D8 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C5 |
| prime radical | C2×C5×D8 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C4×C5×Q8 ◁ G | |
404 conjugacy classes of subgroups; only the 208 normal subgroups are shown.
This group has 404 conjugacy classes of subgroups — too many to draw. Only its 208 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2,3,4,5)(6,7,8,9)(10,13,12,11);(6,10,8,12)(7,11,9,13);(14,15,16,17,18,19,20,21);(15,21)(16,20)(17,19)175 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 19055);