Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#2830 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C24 |
| commutator, Frattini, cosocle | C23 | C24×C5 |
| prime radical | C22×C5×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
408 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)(14,15,18,19)(16,20,21,17)(22,23,26,29)(24,25,28,27);(8,12)(11,13)(16,21)(17,20)(22,24,26,28)(23,27,29,25);(6,7)(8,11)(9,10)(12,13)(14,16)(15,17)(18,21)(19,20)(22,24)(23,27)(25,29)(26,28);(6,8)(7,11)(9,12)(10,13)(14,17)(15,16)(18,20)(19,21)(22,25,26,27)(23,28,29,24)130 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18507);