Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C5×SD64 ⋊ C2 |
| Sylow | C5 ⋊ #2262 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D5×D16 |
| commutator, cosocle | C5×C16 | C23 |
| Frattini | C16 | C22×D5 |
| Fitting | C5×SD64 | C2 |
| socle | C2×C5 | C2×D16 |
| hypercenter | SD64 | D5 |
| hypocenter | C5 | #2262 |
| LCS | G ▷ C5×C16 ▷ C5×C8 ▷ C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ SD64 | |
106 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 106 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 320 |
Polycyclic presentation:
2^7,5|c^2=g,d^2=ef,e^2=fg,f^2=g,b^a=bd,d^a=de,e^a=ef,f^a=fg,h^a=h^4,d^b=de,e^b=ef,f^b=fg,h^c=h^476 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#2262, C2×#10600172, #12257442, … |
| maximal normal subgroup of | D5×#2262, C2×#10600172, #12257442, … |
| maximal quotient of | #12019731, #12019733, #12019802, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 2254);