Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C5×#413 ⋊ C2 |
| Sylow | C5 ⋊ #3018 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #4028 |
| commutator, cosocle | C2×C4×C5 | C24 |
| Frattini | C2×C4 | C23×D5 |
| Fitting | C5×#413 | C2 |
| socle | C2×C5 | C2×#120 |
| hypercenter | #413 | D5 |
| hypocenter | C5 | #3018 |
| LCS | G ▷ C2×C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #413 | |
724 conjugacy classes of subgroups; only the 145 normal subgroups are shown.
This group has 724 conjugacy classes of subgroups — too many to draw. Only its 145 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|d^2=g,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=h^4,e^b=eg,f^b=fg,h^b=h^4,e^c=eg76 conjugacy classes — view the full character table.
| maximal subgroup of | #12016621, #12016626, #12016631, … |
| maximal normal subgroup of | #12016621, #12016626, #12016631, … |
| maximal quotient of | #12071410, #12071461, #12071512, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 12529);