Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C28 |
| direct product | C5×#3504 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C25 |
| commutator, Frattini, cosocle | C22 | C25×C5 |
| prime radical | C2×C5×#41 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ G | |
988 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)(6,7,9,11)(8,12,13,10)(14,15)(16,20)(17,24)(18,19)(21,25)(22,28)(23,26)(27,29);(15,19)(20,26)(21,24)(27,28);(6,8,9,13)(7,10,11,12)(14,16,18,23)(15,20,19,26)(17,22,25,29)(21,27,24,28);(14,15)(16,20)(17,21)(18,19)(22,27)(23,26)(24,25)(28,29);(14,17,18,25)(15,21,19,24)(16,22,23,29)(20,27,26,28)220 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21371);