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×#3509 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C25 |
| commutator, Frattini, cosocle | C22 | C25×C5 |
| prime radical | C2×C5×#41 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ G | |
992 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)(14,15,18,21)(16,19,24,22)(17,25,26,20)(23,28,29,27);(6,7,9,11)(8,10,12,13)(14,15)(16,22)(17,20)(18,21)(19,24)(23,28)(25,26)(27,29);(6,8,9,12)(7,10,11,13);(14,16)(15,19)(17,23)(18,24)(20,27)(21,22)(25,28)(26,29);(14,17,18,26)(15,20,21,25)(16,23,24,29)(19,27,22,28)220 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21376);