Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3585 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C29 |
| direct product | C5×#3585 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C23×D4 |
| commutator, Frattini, cosocle | C4 | C25×C5 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C5×#41 ◁ G | |
1464 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 5 generators on 21 points:
(1,2,3,4,5)(6,7)(8,12)(9,11)(10,13)(14,17)(15,19)(16,18)(20,21);(7,11)(9,17)(12,18)(13,14)(16,21)(19,20);(6,8)(7,12)(9,16)(10,15)(11,18)(13,19)(14,20)(17,21);(8,15)(12,19)(16,21)(18,20);(7,13)(11,14)(12,19)(18,20)205 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21452);