Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | C4⋊(#585⋊C5) |
| unambiguous semidirect product | #63003⋊C5 |
| Fitting, Sylow | #63003 ⋊ C5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C22×C24⋊C5 |
| commutator, cosocle | C2×2−1+4 | C22×C5 |
| Fitting | #63003 | C5 |
| hypercenter | C4⋊C4 | C24⋊C5 |
| hypocenter | 2−1+4 | C5×D4 |
| LCS | G ▷ C2×2−1+4 ▷ 2−1+4 | |
| UCS | C1 ◁ C22 ◁ C4⋊C4 | |
1176 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 1176 conjugacy classes of subgroups — too many to draw. Only its 23 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 256 |
Polycyclic presentation:
5,2^8|b^2=i,c^2=h,e^2=i,f^2=i,b^a=bdei,d^a=dfgi,e^a=dgi,f^a=efghi,g^a=deg,c^b=ch,d^b=di,g^d=gi,f^e=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1116331);