Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5×C8 · Schur multiplier (H2) = C25 |
| direct product | C5×#2988 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C24 |
| commutator | C2 | C23×C5×C8 |
| Frattini, cosocle | C2×C4 | C24×C5 |
| socle | C22×C5 | C23×C4 |
| prime radical | C2×C5×#41 | C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ G | |
432 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2,3,4,5)(7,10)(8,12)(14,15,16,18,17,19,20,21);(6,7)(8,11)(9,10)(12,13);(7,10)(11,13);(6,8,9,12)(7,11,10,13)340 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18665);