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×#2508 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C5 | C24 |
| commutator, Frattini, cosocle | C23 | C24×C5 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
380 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,3,4,5)(6,7,10,8)(9,12,13,11)(15,17)(16,20)(23,27)(25,28);(7,8)(11,12)(14,15,18,17)(16,19,20,21)(22,23)(24,28)(25,29)(26,27);(6,8,10,7)(9,12,13,11)(14,16)(15,19)(17,21)(18,20)(22,24,26,29)(23,28,27,25);(6,9,10,13)(7,11,8,12)(14,17,18,15)(16,21,20,19)(22,25,26,28)(23,29,27,24)190 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18185);