Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#3162 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C5 |
| socle | C22×C5 | C22×D4 |
| prime radical | C23×C5 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C42×C5 ◁ G | |
348 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)(6,7,9,11)(8,12,13,10)(14,15)(16,23)(17,21)(18,27)(19,22)(20,26)(24,29)(25,28);(6,8,9,13)(7,10,11,12)(16,24)(20,28)(23,29)(25,26);(14,16,18,26,19,24,17,25)(15,20,21,23,22,28,27,29);(14,17,19,18)(15,21,22,27)(16,25,24,26)(20,23,28,29)160 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18839);