Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#2950 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D4 |
| commutator | C4 | C23×C4×C5 |
| Frattini, cosocle | C2×C4 | C24×C5 |
| prime radical | C23×C5 | C2×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C5×#117 ◁ G | |
456 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2,3,4,5)(6,7,9,8)(10,11)(12,20)(13,17)(14,26)(15,18)(16,22)(19,28)(21,34)(23,33)(24,32)(25,38)(27,30)(29,36)(31,37)(35,41)(39,40);(10,12,15,23)(11,16,18,30)(13,21,24,35)(14,27,28,22)(17,29,32,40)(19,33,26,20)(25,39,37,36)(31,41,38,34);(6,8,9,7)(10,13,15,24)(11,17,18,32)(12,21,23,35)(14,25,28,37)(16,29,30,40)(19,31,26,38)(20,34,33,41)(22,36,27,39);(11,18)(13,24)(16,30)(19,26)(20,33)(21,35)(25,37)(36,39)220 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18627);