Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #63067⋊C5 |
| Fitting, Sylow | #63067 ⋊ C5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C22×C24⋊C5 |
| commutator, cosocle, hypocenter | #564 | C22×C5 |
| Fitting | #63067 | C5 |
| hypercenter | C2×D4 | C24⋊C5 |
| LCS | G ▷ #564 | |
| UCS | C1 ◁ C22 ◁ C2×D4 | |
676 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 676 conjugacy classes of subgroups — too many to draw. Only its 29 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:
5,2^8|b^2=hi,c^2=h,d^2=hi,e^2=h,b^a=bcdei,c^a=cdh,d^a=bhi,e^a=bcei,c^b=ch,d^b=di,e^b=eh,d^c=dhi,e^c=ei,g^f=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1116333);