Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12307 |
| automorphism | Aut(G) = #72308726 (order 2048) · Inn(G) = #1616 · Out(G) = C24 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1616 |
| commutator, Frattini, cosocle | C2×M16 | C23 |
| prime radical | #3253 | C2 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
159 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 159 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,4,7,5,10,15,9)(3,11,13,8,14,6,12,16);(2,6)(3,12)(4,15)(7,8)(9,16)(10,11)(13,14);(2,7,10,9)(6,8,11,16)Polycyclic presentation:
2^8|a^2=fg,c^2=g,d^2=f,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fh,g^a=gh,d^b=dfh,e^b=egh,f^b=fh,d^c=dg,e^d=eh25 conjugacy classes — view the full character table.
| maximal subgroup of | #151216, #151218, #151252, … |
| maximal normal subgroup of | #151216, #151218, #151252, … |
| maximal quotient of | #126353, #126389, #126450, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5295);