Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#2448 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C24 |
| commutator, Frattini, cosocle | C23 | C24×C5 |
| prime radical | C22×C5×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
512 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)(10,11,14,12)(13,16,17,15)(18,19,22,25)(20,24,21,23);(7,9)(11,12)(15,16)(18,20,22,21)(19,23,25,24);(6,7)(8,9)(10,12,14,11)(13,16,17,15)(18,19)(20,23)(21,24)(22,25);(10,13,14,17)(11,15,12,16)(18,21,22,20)(19,24,25,23)190 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18125);