Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C5×#408 ⋊ C2 |
| Sylow | C5 ⋊ #3198 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×D20 |
| commutator, cosocle | C2×C4×C5 | C24 |
| Frattini | C2×C4 | C23×D5 |
| Fitting | C5×#408 | C2 |
| socle | C22×C5 | C22×D4 |
| hypercenter | #408 | D5 |
| hypocenter | C5 | #3198 |
| LCS | G ▷ C2×C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C2×#41 ◁ #408 | |
422 conjugacy classes of subgroups.
| transitive degree | μt(G) = 160 |
Polycyclic presentation:
2^7,5|f^2=g,b^a=be,c^a=cf,d^a=de,f^a=fg,h^a=h^4,c^b=ce,d^b=dg,f^c=fg,h^c=h^482 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#3198, C2×#10610404, #12970024, … |
| maximal normal subgroup of | D5×#3198, C2×#10610404, #12970024, … |
| maximal quotient of | #12108399, #12108419, #12109601, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 12486);