Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C23×C4 |
| direct product | C5×#3136 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C5 |
| prime radical | C5×#535 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C5×#117 ◁ G | |
404 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,3,4,5)(6,7)(8,11)(9,10)(12,13)(15,19)(16,22)(17,24)(20,27)(21,26)(23,28);(7,10)(11,13)(14,15,17,19,18,21,24,26)(16,20,22,25,23,27,28,29);(6,8,9,12)(7,11,10,13)(14,16,17,22,18,23,24,28)(15,20,19,25,21,27,26,29);(15,21)(19,26)(20,27)(25,29)160 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18813);