Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1406 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C4 |
| direct product | C5×#1407 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #353 |
| commutator | C42 | C2×C4×C5 |
| Frattini, cosocle | #124 | C22×C5 |
| prime radical | C2×C5×D4 | D4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C5 ◁ C2×C4×C5 ◁ C5×#124 ◁ G | |
98 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)(8,16,18,31)(9,20,21,36)(10,23,25,14)(11,15)(12,17,26,33)(13,22)(19,29,24,32)(27,34,28,35)(30,37);(6,8,11,19)(7,12,15,28)(9,17,21,35)(10,24,25,18)(13,29,30,16)(14,26,23,27)(20,32,36,31)(22,33,37,34)70 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 540);