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×#3407 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C8 | C24 |
| commutator | C2 | C23×C5×C8 |
| Frattini, cosocle | C8 | C24×C5 |
| socle | C2×C5 | C23×C8 |
| prime radical | C2×C5×#41 | C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C5×C8 ◁ G | |
360 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7,9,14,10,15,19,27,11,13,20,25,21,26,31,35)(8,12,17,23,18,24,30,34,16,22,28,32,29,33,36,37);(6,8)(7,12)(9,17)(10,18)(11,16)(13,22)(14,23)(15,24)(19,30)(20,28)(21,29)(25,32)(26,33)(27,34)(31,36)(35,37);(8,16)(12,22)(17,28)(18,29)(23,32)(24,33)(30,36)(34,37);(7,13)(12,22)(14,25)(15,26)(23,32)(24,33)(27,35)(34,37)340 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 19084);