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×#2744 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C24 |
| commutator, Frattini, cosocle | C23 | C24×C5 |
| prime radical | C24×C5 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
392 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,3,4,5)(6,7,9,10)(8,11,12,13)(15,16)(19,20)(22,23)(24,27)(25,29)(26,28);(7,10)(11,13)(14,15,18,16)(17,20,21,19)(22,24,26,29)(23,27,28,25);(6,8,9,12)(7,11,10,13)(14,16,18,15)(17,20,21,19)(22,24)(23,25)(26,29)(27,28);(6,7)(8,11)(9,10)(12,13)(14,17)(15,19)(16,20)(18,21)(22,25)(23,24)(26,27)(28,29)160 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18421);