Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C5×#458 ⋊ C2 |
| Sylow | C5 ⋊ #3026 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D5×#120 |
| commutator, cosocle | C23×C5 | C24 |
| Frattini | C23 | C23×D5 |
| Fitting | C5×#458 | C2 |
| socle | C2×C5 | C2×#120 |
| hypercenter | #458 | D5 |
| hypocenter | C5 | #3026 |
| LCS | G ▷ C23×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ #458 | |
800 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 800 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 80 |
Polycyclic presentation:
2^7,5|b^a=be,c^a=cf,d^a=dg,h^a=h^4,f^b=fg,e^c=eg,h^d=h^461 conjugacy classes — view the full character table.
| maximal subgroup of | #12253358, #12253360, #12253369, … |
| maximal normal subgroup of | #12253358, #12253360, #12253369, … |
| maximal quotient of | #12073135, #12073142, #12073149, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 14124);