Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#2831 |
| 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 | |
366 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)(17,19,21,20)(22,23,26,27)(24,28,29,25);(8,12)(11,13)(15,16)(17,21)(24,29)(25,28);(6,7)(8,11)(9,10)(12,13)(14,16,18,15)(17,19,21,20)(22,24)(23,25)(26,29)(27,28);(6,8)(7,11)(9,12)(10,13)(14,17,18,21)(15,19,16,20)(22,25)(23,24)(26,28)(27,29)130 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18508);