Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C5 ⋊ #2168 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C8 | C22×D5 |
| commutator, socle | C22×C5 | C22×C8 |
| Fitting | C5×C8×D4 | C2 |
| prime radical | C2×C5×D4 | C8 |
| cosocle | C2×C5×C8 | C23 |
| hypercenter | C8×D4 | D5 |
| hypocenter | C5 | #2168 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C2×C8 ◁ C8×D4 | |
164 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 164 conjugacy classes of subgroups — too many to draw. Only its 87 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 320 |
Polycyclic presentation:
2^7,5|a^2=e,c^2=e,e^2=f,f^2=g,b^a=bd,c^a=cg,h^a=h^4,h^c=h^4136 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#2168, C2×#10601650, #12252466, … |
| maximal normal subgroup of | D5×#2168, C2×#10601650, #12252466, … |
| maximal quotient of | #12011011, #12011074, #12011081, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 3732);