Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1304 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C5 · Schur multiplier (H2) = C2 |
| direct product | C5×#1308 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #95 |
| commutator | C23 | C42×C5 |
| Frattini, cosocle | C2×C42 | C22×C5 |
| prime radical | C23×C5 | C42 |
| LCS | G ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
122 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 2 generators on 37 points:
(1,2,3,4,5)(6,7,9,14,11,13,10,15)(8,16,17,21,18,12,19,20)(22,23,25,28,26,29,33,36)(24,30,32,37,31,27,34,35);(6,8,10,17,11,18,9,19)(7,12,15,20,13,16,14,21)(22,24)(23,27,29,30)(25,32)(26,31)(28,35,36,37)(33,34)130 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 441);