Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#2802 |
| 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 | |
360 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)(7,10)(8,12)(14,15,17,19)(16,20,21,18)(22,23)(24,28)(25,27)(26,29);(7,10)(11,13)(22,23,25,27)(24,26,29,28);(6,7,9,10)(8,11,12,13)(14,15)(16,18)(17,19)(20,21)(22,24)(23,26)(25,29)(27,28);(6,8,9,12)(7,11,10,13)(14,16,17,21)(15,18,19,20)(22,23)(24,26)(25,27)(28,29)160 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18479);