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×#2949 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D4 |
| commutator | C4 | C23×C4×C5 |
| Frattini, cosocle | C2×C4 | C24×C5 |
| prime radical | C22×C5×D4 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C5×#117 ◁ G | |
492 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)(6,7,9,8)(10,11)(12,18)(13,21)(14,16)(15,19)(17,23)(20,25)(22,24);(10,12,14,20)(11,15,16,24)(13,22,23,19)(17,25,21,18);(6,8,9,7)(10,11)(12,15)(13,17)(14,16)(18,19)(20,24)(21,23)(22,25);(11,16)(15,24)(17,21)(18,25)220 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18626);