Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#2833 |
| 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 | |
332 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(7,10)(11,13)(14,15,18,19)(16,20,21,17)(23,27)(28,30)(29,33)(36,37);(8,12)(11,13)(16,21)(17,20)(22,23,26,27)(24,30,32,28)(25,33,35,29)(31,36,34,37);(6,7)(8,11)(9,10)(12,13)(14,16)(15,17)(18,21)(19,20)(22,24,26,32)(23,28,27,30)(25,34,35,31)(29,37,33,36);(6,8)(7,11)(9,12)(10,13)(14,17)(15,16)(18,20)(19,21)(22,25,26,35)(23,29,27,33)(24,31,32,34)(28,36,30,37)130 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18510);