Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C5 ⋊ #2399 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D5 |
| commutator, socle, prime radical, cosocle | C23×C5 | C24 |
| Frattini | C23 | C23×D5 |
| Fitting | C2×C4×C5×Q8 | C2 |
| hypercenter | C2×C4×Q8 | D5 |
| hypocenter | C5 | #2399 |
| LCS | G ▷ C23×C5 ▷ C5 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×Q8 | |
504 conjugacy classes of subgroups; only the 177 normal subgroups are shown.
This group has 504 conjugacy classes of subgroups — too many to draw. Only its 177 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 640 |
Polycyclic presentation:
2^7,5|a^2=e,b^2=e,c^2=e,d^2=ef,b^a=be,c^a=cf,c^b=cg,h^b=h^4124 conjugacy classes — view the full character table.
| maximal subgroup of | #11994371, #11994372, #11994374, … |
| maximal normal subgroup of | #11994371, #11994372, #11994374, … |
| maximal quotient of | #12021041, #12021056, #12021057, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 16280);