Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#2527 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C24 |
| commutator, Frattini, cosocle | C23 | C24×C5 |
| prime radical | C24×C5 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
420 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,3,4,5)(6,7,9,11)(8,10,12,13)(14,15)(16,20)(17,19)(18,21)(24,28)(27,29);(6,7,9,11)(8,10,12,13)(14,15,17,19)(16,18,21,20)(22,23,26,25)(24,27,28,29);(6,7)(8,10)(9,11)(12,13)(14,15)(16,18)(17,19)(20,21)(22,24)(23,27)(25,29)(26,28);(6,8,9,12)(7,10,11,13)(14,16)(15,18)(17,21)(19,20)(22,25,26,23)(24,29,28,27)190 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18204);