Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#2931 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C8 | C23 |
| commutator | C22 | C23×C4×C5 |
| Frattini, cosocle | C2×C4 | C24×C5 |
| socle | C22×C5 | C23×C4 |
| prime radical | C5×#532 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C8 ◁ G | |
500 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)(8,9)(10,11,15,18,14,17,23,25)(12,19,21,22,20,16,24,13);(7,9)(10,12)(11,16)(13,18)(14,20)(15,21)(17,19)(22,25)(23,24);(10,13,15,19,14,22,23,16)(11,12,18,21,17,20,25,24);(11,17)(12,20)(18,25)(21,24)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18608);