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×#2505 |
| 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 | |
456 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)(6,7,10,11)(8,13,12,9)(14,15,18,20)(16,17,21,19)(23,26)(24,28);(7,11)(8,12)(14,16,18,21)(15,19,20,17)(22,23)(24,27)(25,26)(28,29);(6,8,10,12)(7,9,11,13)(14,16)(15,17)(18,21)(19,20)(22,24,25,28)(23,27,26,29);(6,9)(7,12)(8,11)(10,13)(14,17)(15,16)(18,19)(20,21)(22,23)(24,27)(25,26)(28,29)190 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18182);