Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#3054 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C5 |
| socle | C22×C5 | C22×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
500 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2,3,4,5)(6,7)(8,9)(10,11)(12,19)(13,17)(14,23)(15,18)(16,22)(20,25)(21,24);(10,12,14,22,15,20,13,21)(11,16,17,19,18,24,23,25);(7,9)(12,20)(16,24)(19,25)(21,22);(10,13,15,14)(11,17,18,23)(12,21,20,22)(16,19,24,25)160 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18731);