Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C5 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #3573⋊C5 |
| Fitting, Sylow | #3573 ⋊ C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C8 | C24⋊C5 |
| commutator, hypocenter | 2−1+4 | C4×C5 |
| Frattini | C4 | C2×C24⋊C5 |
| Fitting | #3573 | C5 |
| socle | C2 | C4×C24⋊C5 |
| prime radical | #585⋊C5 | C2 |
| cosocle | #585 | C2×C5 |
| LCS | G ▷ 2−1+4 | |
| UCS | C1 ◁ C8 | |
155 conjugacy classes of subgroups.
| transitive degree | μt(G) = 128 |
Polycyclic presentation:
2,5,2^6|a^2=c,c^2=h,d^2=h,f^2=h,g^2=h,d^b=defh,e^b=efh,f^b=degh,g^b=dgh,g^d=gh,f^e=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #62624⋊C5, C2×#3573⋊C5, #63076⋊C5, … |
| maximal normal subgroup of | #62624⋊C5, C2×#3573⋊C5, #63076⋊C5, … |
| maximal quotient of | C8×2−1+4⋊C5, #60332⋊C5, C2×#3573⋊C5, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 19104);