Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C2×C5×#413 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C22⋊C4 |
| commutator | C22 | C23×C4×C5 |
| Frattini, cosocle | C2×C4 | C24×C5 |
| socle | C22×C5 | C2×C22⋊C4 |
| prime radical | C22×C5×#41 | C2 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
756 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2)(19,20,21,22,23);(3,4,7,12,8,11,6,10)(5,13,14,18,15,9,16,17);(3,5)(4,9)(6,14)(7,16)(8,15)(10,17)(11,13)(12,18);(3,6,8,7)(4,10,11,12)(5,14,15,16)(9,17,13,18)220 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18564);