Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C5×#3293 ⋊ C2 |
| Sylow | C5 ⋊ #61105 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D42×D5 |
| commutator, cosocle | C2×C4×C5 | C25 |
| Frattini | C2×C4 | C24×D5 |
| Fitting | C5×#3293 | C2 |
| socle | C2×C5 | C2×D42 |
| hypercenter | #3293 | D5 |
| hypocenter | C5 | #61105 |
| LCS | G ▷ C2×C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #3293 | |
2620 conjugacy classes of subgroups.
| transitive degree | μt(G) = 160 |
Polycyclic presentation:
2^8,5|e^2=h,f^2=h,g^2=h,b^a=bf,c^a=cg,f^a=fh,g^a=gh,i^a=i^4,c^b=ch,d^b=df,f^b=fh,g^c=gh,f^d=fh,i^d=i^4,i^e=i^4116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1045808);