Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C23×C4 |
| direct product | C5×#2331 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C5 | C24 |
| commutator | C22 | C23×C4×C5 |
| Frattini, cosocle | C23 | C24×C5 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
460 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2,3,4,5)(6,7,9,8)(10,11,13,14)(12,16,17,15)(20,24)(22,25);(6,8,9,7)(11,14)(15,16)(18,19,21,23)(20,22,24,25);(10,11)(12,15)(13,14)(16,17)(18,20,21,24)(19,22,23,25);(10,12,13,17)(11,15,14,16)220 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18008);