Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C28 |
| direct product | C5×#3502 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C25 |
| commutator, Frattini, cosocle | C22 | C25×C5 |
| prime radical | C23×C5 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ G | |
964 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 5 generators on 29 points:
(1,2,3,4,5)(6,7,9,11)(8,12,13,10)(14,15,19,23)(16,21,17,20)(18,22,26,29)(24,28,25,27);(14,16,19,17)(15,20,23,21)(18,25,26,24)(22,28,29,27);(6,8,9,13)(7,10,11,12)(14,17,19,16)(15,21,23,20)(18,25,26,24)(22,28,29,27);(14,18,19,26)(15,22,23,29)(16,24,17,25)(20,27,21,28);(14,15)(16,20)(17,21)(18,22)(19,23)(24,27)(25,28)(26,29)220 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21369);