Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C5 · Schur multiplier (H2) = C29 |
| direct product | C5×#3573 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C8 | C24 |
| commutator | C2 | C24×C4×C5 |
| Frattini, cosocle | C4 | C25×C5 |
| socle | C2×C5 | C24×C4 |
| prime radical | C5×#585 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C5×C8 ◁ G | |
1424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 5 generators on 37 points:
(1,2,3,4,5)(6,7,10,14,11,15,21,27)(8,12,18,24,17,23,30,34)(9,13,20,26,19,25,31,35)(16,22,29,33,28,32,36,37);(6,8)(7,12)(9,16)(10,18)(11,17)(13,22)(14,24)(15,23)(19,28)(20,29)(21,30)(25,32)(26,33)(27,34)(31,36)(35,37);(6,9)(7,13)(8,16)(10,20)(11,19)(12,22)(14,26)(15,25)(17,28)(18,29)(21,31)(23,32)(24,33)(27,35)(30,36)(34,37);(9,19)(13,25)(16,28)(20,31)(22,32)(26,35)(29,36)(33,37);(8,17)(12,23)(16,28)(18,30)(22,32)(24,34)(29,36)(33,37)340 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21440);