Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#2769 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C5 | C24 |
| commutator, Frattini, cosocle | C23 | C24×C5 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
344 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,3,4,5)(6,7,10,8)(9,12,13,11)(14,15)(16,19)(17,18)(20,21)(24,27)(25,29);(7,8)(9,13)(15,17)(19,20)(22,23,26,28)(24,25,27,29);(6,8,10,7)(9,12,13,11)(14,16,18,21)(15,19,17,20)(22,24,26,27)(23,25,28,29);(6,9,10,13)(7,11,8,12)(14,17,18,15)(16,20,21,19)(22,25)(23,27)(24,28)(26,29)160 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18446);