Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1357 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C2 |
| direct product | C5×#1358 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #331 |
| commutator | C42 | C2×C4×C5 |
| Frattini, cosocle | C4×C8 | C22×C5 |
| prime radical | C22×C5 | #100 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C4×C5 ◁ C4×C5×C8 ◁ G | |
92 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 2 generators on 37 points:
(1,2,3,4,5)(6,7,9,14,10,13,22,29,11,16,23,32,25,15,33,31)(8,17,18,34,19,28,35,37,20,27,36,30,24,12,21,26);(6,8,11,20)(7,12,16,28)(9,21,23,35)(10,24,25,19)(13,27,15,17)(14,30,32,34)(18,22,36,33)(26,29,37,31)115 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 491);