Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C24 |
| unambiguous semidirect product | #63103⋊C5 |
| Fitting, Sylow | #63103 ⋊ C5 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×C24⋊C5 |
| commutator, cosocle, hypocenter | 2−1+4 | C23×C5 |
| Frattini, socle | C2 | C23×C24⋊C5 |
| Fitting | #63103 | C5 |
| hypercenter | #41 | C24⋊C5 |
| LCS | G ▷ 2−1+4 | |
| UCS | C1 ◁ C4 ◁ #41 | |
6010 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 6010 conjugacy classes of subgroups — too many to draw. Only its 49 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
Polycyclic presentation:
5,2^8|h^2=i,b^a=dfgi,c^a=cg,d^a=dg,e^a=bcdgi,f^a=fg,g^a=bceh,c^b=ci,g^b=gi,f^c=fi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1116355);