Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5×C8 · Schur multiplier (H2) = C2 |
| direct product | C5×#1327 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #99 |
| commutator | C2×C4 | C2×C5×C8 |
| Frattini, cosocle | C2×M16 | C22×C5 |
| socle | C2×C5 | #324 |
| prime radical | C23×C5 | M16 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C2×C4×C5 ◁ C2×C5×M16 ◁ G | |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,2,3,4,5)(6,7,9,14,10,13,18,20,11,15,17,21,8,12,16,19);(7,12,15,13)(9,17)(14,19,21,20)(16,18)130 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 460);