Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C5×#504 ⋊ C2 |
| Sylow | C5 ⋊ #3416 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×D40 |
| commutator, cosocle | C5×C8 | C24 |
| Frattini | C8 | C23×D5 |
| Fitting | C5×#504 | C2 |
| socle | C2×C5 | C22×D8 |
| hypercenter | #504 | D5 |
| hypocenter | C5 | #3416 |
| LCS | G ▷ C5×C8 ▷ C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ #41 ◁ #131 ◁ #504 | |
360 conjugacy classes of subgroups; only the 129 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 129 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 160 |
Polycyclic presentation:
2^7,5|a^2=g,e^2=fg,f^2=g,b^a=be,e^a=ef,f^a=fg,h^a=h^4,e^b=ef,f^b=fg,h^b=h^4,d^c=dg112 conjugacy classes — view the full character table.
| maximal subgroup of | #12256051, #12256053, #12256058, … |
| maximal normal subgroup of | #12256051, #12256053, #12256058, … |
| maximal quotient of | #12252213, #12252220, #12252349, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 15827);