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×#2789 |
| 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 | |
352 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)(8,12)(9,10)(11,13)(14,15,19,21)(16,18,17,20)(22,23,26,28)(24,25,27,29);(7,10)(11,12)(14,16,19,17)(15,20,21,18)(22,24)(23,25)(26,27)(28,29);(6,8,9,13)(7,11,10,12)(14,17,19,16)(15,20,21,18)(22,25)(23,27)(24,28)(26,29);(6,7)(8,11)(9,10)(12,13)(14,18,19,20)(15,16,21,17)160 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18466);