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×#3507 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C25 |
| commutator, Frattini, cosocle | C22 | C25×C5 |
| prime radical | C5×#532 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ G | |
1012 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)(8,12)(10,13)(14,15)(16,23)(17,18)(19,20)(21,26)(22,27)(24,29)(25,28);(6,7,9,11)(8,10,12,13)(15,20)(17,22)(21,23)(24,28);(6,8,9,12)(7,10,11,13)(14,16,19,26)(15,21,20,23)(17,24,22,28)(18,25,27,29);(14,17,19,22)(15,18,20,27)(16,24,26,28)(21,25,23,29);(14,18)(15,22)(16,25)(17,20)(19,27)(21,28)(23,24)(26,29)220 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21374);