Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C5×#2196 ⋊ C2 |
| Sylow | C5 ⋊ #13674 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #10601789 |
| commutator, cosocle | C5×#127 | C23 |
| Frattini | #127 | C22×D5 |
| Fitting | C5×#2196 | C2 |
| socle | C2×C5 | #2196 |
| hypercenter | #2196 | D5 |
| hypocenter | C5 | #13674 |
| LCS | G ▷ C5×#127 ▷ C2×C4×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #127 ◁ #2196 | |
454 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 454 conjugacy classes of subgroups — too many to draw. Only its 43 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 80 |
Polycyclic presentation:
2^8,5|b^2=h,f^2=h,b^a=bd,c^a=ce,f^a=fg,e^b=efgh,g^b=gh,d^c=df,f^c=fh,g^c=gh,i^c=i^4,e^d=egh,f^d=fh,f^e=fh65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 90282);