Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#2509 |
| 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)(7,10)(8,12)(14,15,18,21)(16,20,17,19)(22,23,27,25)(24,29,28,26);(6,7,9,10)(8,11,12,13)(14,16,18,17)(15,19,21,20)(22,24)(23,26)(25,29)(27,28);(6,7)(8,11)(9,10)(12,13)(14,15)(16,19)(17,20)(18,21)(22,25,27,23)(24,29,28,26);(6,8,9,12)(7,11,10,13)(14,17,18,16)(15,20,21,19)(22,26,27,29)(23,28,25,24)190 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18186);