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×#2767 |
| 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)(6,7,10,9)(8,12,13,11)(16,19)(17,21)(23,26)(24,28);(7,9)(8,13)(14,15,18,20)(16,17,19,21)(22,23,25,26)(24,29,28,27);(6,8,10,13)(7,11,9,12)(14,16,18,19)(15,17,20,21);(6,9,10,7)(8,12,13,11)(14,17)(15,19)(16,20)(18,21)(22,24,25,28)(23,27,26,29)160 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18444);