Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| Fitting | C5×2−1+6 ⋊ C2 |
| Sylow | C5 ⋊ #60400 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×#4099 |
| commutator, cosocle | C23×C5 | C25 |
| Frattini | C23 | C24×D5 |
| Fitting | C5×2−1+6 | C2 |
| socle | C2×C5 | C22×#120 |
| hypercenter | 2−1+6 | D5 |
| hypocenter | C5 | #60400 |
| LCS | G ▷ C23×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ C22×Q8 ◁ 2−1+6 | |
4732 conjugacy classes of subgroups; only the 807 normal subgroups are shown.
This group has 4732 conjugacy classes of subgroups — too many to draw. Only its 807 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 160 |
Polycyclic presentation:
2^8,5|d^2=h,e^2=h,b^a=bf,c^a=cg,i^a=i^4,g^b=gh,f^c=fh,e^d=eh188 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1036325);