Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1655 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#1658 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #120 |
| commutator, Frattini, cosocle | C42 | C23×C5 |
| prime radical | C5×#127 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C42×C5 ◁ G | |
182 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(7,12)(8,15)(9,10)(14,17)(16,18)(19,20)(23,27)(25,32)(28,36)(29,33)(30,34)(35,37);(6,7)(8,13)(9,17)(10,12)(11,14)(15,20)(16,21)(18,19)(22,23,26,29)(24,28,31,37)(25,33,32,27)(30,35,34,36);(6,8,11,16)(7,13,14,21)(9,15,10,18)(12,19,17,20)(22,24,26,31)(23,28,29,37)(25,34,32,30)(27,35,33,36)100 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 5248);