Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3192 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#3235 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C5 |
| prime radical | C5×#532 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
366 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)(8,11)(12,13)(14,15)(16,22)(17,20)(18,21)(19,24)(23,28)(25,27)(26,29);(6,7,9,10)(8,11,12,13)(14,16,18,24)(15,19,21,22)(17,25,26,23)(20,28,29,27);(6,8,9,12)(7,11,10,13)(14,17,18,26)(15,20,21,29)(16,23,24,25)(19,27,22,28);(6,7,9,10)(8,11,12,13)(15,21)(19,22)(20,29)(27,28)130 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18912);