Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C2×C5×#400 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C5 | C23 |
| commutator, Frattini, cosocle | C23 | C24×C5 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C5 ◁ G | |
604 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2)(23,24,25,26,27);(9,13)(11,14)(15,16,18,20)(17,21,22,19);(4,6)(7,8,10,12)(9,14,13,11)(15,16)(17,21)(18,20)(19,22);(3,4)(5,6)(7,9)(8,11)(10,13)(12,14)(15,17,18,22)(16,19,20,21)220 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18071);