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×#2774 |
| 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 | |
368 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)(6,7,10,9)(8,12,13,11)(14,15)(16,20)(17,19)(18,21)(22,23,26,28)(24,25,29,27);(7,9)(8,13)(14,15,17,19)(16,20,21,18)(22,24,26,29)(23,27,28,25);(6,8,10,13)(7,11,9,12)(14,15)(16,18)(17,19)(20,21);(6,9,10,7)(8,12,13,11)(14,16,17,21)(15,18,19,20)(22,25,26,27)(23,24,28,29)160 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18451);