Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C22×C5×D16 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | D8 |
| commutator, Frattini, cosocle | C8 | C24×C5 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C22×C4×C5 ◁ C22×C5×C8 ◁ G | |
440 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)(21,22,23,24,25);(3,4);(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20);(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)220 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 19085);