Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C2×C5×#443 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C8 | D4 |
| commutator | C4 | C23×C4×C5 |
| Frattini, cosocle | C2×C4 | C24×C5 |
| socle | C22×C5 | C4×D4 |
| prime radical | C2×C5×#135 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C8 ◁ C22×C5×C8 ◁ G | |
472 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2)(19,20,21,22,23);(3,4,7,11,8,5,12,10)(6,9,14,17,15,13,18,16);(3,5)(4,8)(6,13)(7,10)(9,15)(11,12)(14,16)(17,18);(3,6,7,14,8,15,12,18)(4,9,11,17,5,13,10,16)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18630);