Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C4 |
| direct product | C5×#2244 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23×C5 |
| prime radical | C5×#493 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C42×C5 ◁ C4×C5×C8 ◁ G | |
152 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,3,4,5)(6,7)(8,11)(9,10)(12,13)(15,21)(16,27)(17,22)(18,29)(19,23)(24,25)(26,28);(7,10)(11,13)(14,15,22,16,19,25,29,28,20,26,18,24,23,27,17,21);(6,8,9,12)(7,11,10,13)(14,16,17,15,23,21,18,27,20,24,29,26,19,28,22,25)130 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 5834);