Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C28 |
| direct product | C5×#3562 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C25 |
| commutator, Frattini, cosocle | C22 | C25×C5 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ G | |
1050 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 5 generators on 29 points:
(1,2,3,4,5)(7,10)(8,12)(14,15)(16,19)(17,22)(18,23)(20,24)(21,26)(25,27)(28,29);(7,10)(11,13)(15,16)(17,24)(21,23)(25,29);(6,7)(8,11)(9,10)(12,13)(14,16,19,15)(17,22,24,20)(18,23,26,21)(25,28,29,27);(6,8,9,12)(7,11,10,13)(14,17)(15,20)(16,22)(18,25)(19,24)(21,27)(23,28)(26,29);(14,18,19,26)(15,21,16,23)(17,25,24,29)(20,27,22,28)190 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21429);