Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#2380 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C24 |
| commutator | C22 | C23×C4×C5 |
| Frattini, cosocle | C23 | C24×C5 |
| prime radical | C2×C5×#127 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
648 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)(11,14)(12,16)(18,19,22,20)(21,24,25,23);(11,14)(15,17)(19,20)(21,25);(10,11,13,14)(12,15,16,17)(18,20,22,19)(21,24,25,23);(10,12)(11,15)(13,16)(14,17)(18,21,22,25)(19,23,20,24)190 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18057);