Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C5×#2411 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C24 |
| commutator, Frattini, cosocle | C23 | C24×C5 |
| prime radical | C2×C5×#120 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
786 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2,3,4,5)(10,11,12,13)(14,15)(16,20)(17,19)(18,21);(7,9)(14,15,17,19)(16,18,21,20);(6,7)(8,9)(10,11)(12,13)(14,15)(16,18)(17,19)(20,21);(14,16)(15,18)(17,21)(19,20)190 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18088);