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×#3461 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C24 |
| commutator, Frattini, cosocle | C22 | C25×C5 |
| socle | C22×C5 | C25 |
| prime radical | C5×#537 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ G | |
1012 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 5 generators on 29 points:
(1,2,3,4,5)(6,7)(8,11)(9,10)(12,13)(14,15)(16,22)(17,20)(18,21)(19,24)(23,28)(25,27)(26,29);(14,16,18,24)(15,19,21,22)(17,25,26,23)(20,28,29,27);(7,10)(11,13)(14,17,18,26)(15,20,21,29)(16,23,24,25)(19,27,22,28);(15,21)(19,22)(20,29)(27,28);(6,8,9,12)(7,11,10,13)220 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21328);