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×#2517 |
| 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 | |
412 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)(7,10)(11,13)(14,15,18,16)(17,20,21,19)(22,23,26,25)(24,28,29,27);(15,16)(19,20)(24,29)(27,28);(6,7)(8,11)(9,10)(12,13)(14,16,18,15)(17,20,21,19)(22,24)(23,27)(25,28)(26,29);(6,8,9,12)(7,11,10,13)(14,17)(15,19)(16,20)(18,21)(22,25,26,23)(24,28,29,27)190 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18194);