Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | C22⋊#66244 |
| Fitting | C22×2−1+4 ⋊ D5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C2×(C24⋊C5)⋊C2 |
| commutator, cosocle | C2×2−1+4⋊C5 | C22 |
| Fitting | C22×2−1+4 | D5 |
| hypercenter | C23 | (C24⋊C5)⋊C2 |
| hypocenter | 2−1+4⋊C5 | D4 |
| LCS | G ▷ C2×2−1+4⋊C5 ▷ 2−1+4⋊C5 | |
| UCS | C1 ◁ C22 ◁ C23 | |
672 conjugacy classes of subgroups; only the 17 normal subgroups are shown.
This group has 672 conjugacy classes of subgroups — too many to draw. Only its 17 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 128 |
Polycyclic presentation:
2,5,2^7|d^2=i,e^2=i,b^a=b^4,c^a=cdef,d^a=df,e^a=efi,g^a=gh,c^b=ce,d^b=dfi,e^b=cef,f^b=cd,d^c=di,e^c=ei,f^c=fi,e^d=ei41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1116419);