Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#3082 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C5 |
| prime radical | C23×C5 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C5×#117 ◁ G | |
362 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,3,4,5)(6,7,9,11)(8,12,13,10)(16,22)(17,24)(19,26)(21,28)(23,25)(27,29);(6,8,9,13)(7,10,11,12)(14,15)(16,19)(17,21)(18,20)(22,26)(23,27)(24,28)(25,29);(14,16,18,23)(15,19,20,27)(17,25,24,22)(21,29,28,26);(15,20)(19,27)(21,28)(26,29)160 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18759);