Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C28 |
| direct product | C5×#3458 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C24 |
| commutator, Frattini, cosocle | C22 | C25×C5 |
| socle | C22×C5 | C25 |
| prime radical | C24×C5 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ G | |
996 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 5 generators on 29 points:
(1,2,3,4,5)(6,7,9,11)(8,12,13,10)(14,15)(16,22)(17,21)(18,20)(19,23)(24,28)(25,27)(26,29);(14,16)(15,19)(17,24)(18,23)(20,22)(21,26)(25,29)(27,28);(6,8,9,13)(7,10,11,12)(16,23)(19,22)(24,29)(26,28);(15,20)(19,22)(21,27)(26,28);(14,17,18,25)(15,21,20,27)(16,24,23,29)(19,26,22,28)220 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21325);