Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#3035 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C5 |
| socle | C22×C5 | C22×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C42×C5 ◁ G | |
496 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2,3,4,5)(7,9)(10,11)(12,19)(13,17)(14,22)(15,18)(16,20)(21,24)(23,25);(6,7,8,9);(10,12)(11,16)(13,20)(14,23)(15,21)(17,24)(18,25)(19,22);(10,13,15,14)(11,17,18,22)(12,20,21,23)(16,24,25,19)190 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18712);