Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C16 · Schur multiplier (H2) = C4 |
No aliases for this group.
| Sylow | C5 ⋊ #1312 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4×C8 | C2×D5 |
| commutator | C2×C5 | C4×C16 |
| Fitting | C5×C82 | C2 |
| socle, prime radical | C22×C5 | C4×C8 |
| cosocle | C4×C5×C8 | C22 |
| hypercenter | C82 | D5 |
| hypocenter | C5 | #1312 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C4×C8 ◁ C82 | |
92 conjugacy classes of subgroups.
| transitive degree | μt(G) = 640 |
Polycyclic presentation:
2^7,5|a^2=c,b^2=d,c^2=e,d^2=f,f^2=g,b^a=be,h^b=h^4208 conjugacy classes — view the full character table.
| maximal subgroup of | #11985916, #11985917, #11985954, … |
| maximal normal subgroup of | #11985916, #11985917, #11985954, … |
| maximal quotient of | #11985916, #11986128, #11986631, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 11);