Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C5×#497 ⋊ C2 |
| Sylow | C5 ⋊ #3289 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #66051 |
| commutator, cosocle | C2×C4×C5 | C24 |
| Frattini | C2×C4 | C23×D5 |
| Fitting | C5×#497 | C2 |
| socle | C2×C5 | D42 |
| hypercenter | #497 | D5 |
| hypocenter | C5 | #3289 |
| LCS | G ▷ C2×C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #497 | |
448 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 448 conjugacy classes of subgroups — too many to draw. Only its 117 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,b^2=g,e^2=g,f^2=g,b^a=be,c^a=cf,e^a=eg,f^a=fg,d^b=de,e^b=eg,h^b=h^4,f^c=fg,h^c=h^4,e^d=eg,h^d=h^458 conjugacy classes — view the full character table.
| maximal subgroup of | #12020038, #12020042, #12020052, … |
| maximal normal subgroup of | #12020038, #12020042, #12020052, … |
| maximal quotient of | #12088676, #12088691, #12088721, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 15698);