Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#2951 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D4 |
| commutator | C4 | C23×C4×C5 |
| Frattini, cosocle | C2×C4 | C24×C5 |
| prime radical | C2×C5×#135 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C5×#117 ◁ G | |
484 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2,3,4,5)(6,7,9,8)(10,11)(12,17)(13,19)(14,18)(15,20)(16,21)(22,25)(23,24);(10,12)(11,16)(13,21)(14,23)(15,22)(17,18)(19,25)(20,24);(6,8,9,7)(10,12)(11,17)(13,23)(14,21)(15,22)(16,18)(19,24)(20,25);(10,13,15,14)(11,18,20,19)(12,21,22,23)(16,17,24,25)220 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18628);