Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(7,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C5 · Schur multiplier (H2) = C214 |
| direct product | C5×H(7,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C26 |
| commutator, Frattini, cosocle | C2 | C26×C5 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ G | |
5992 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 6 generators on 21 points:
(1,2,3,4,5)(6,7)(8,12)(9,16)(10,11)(13,17)(14,18)(15,21)(19,20);(7,11)(12,18)(13,16)(19,21);(8,14)(12,18)(15,20)(19,21);(6,8)(7,12)(9,15)(10,14)(11,18)(13,19)(16,21)(17,20);(6,7)(8,12)(9,13)(10,11)(14,18)(15,19)(16,17)(20,21);(6,9)(7,13)(8,15)(10,17)(11,16)(12,19)(14,20)(18,21)325 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21534);