Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13673 |
| automorphism | Aut(G) = C2×#512862 (order 1024) · Inn(G) = #2196 · Out(G) = C23 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2196 |
| commutator, Frattini, cosocle | #127 | C23 |
| prime radical | #3292 | C2 |
| LCS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #127 ◁ G | |
| LECS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
204 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 204 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.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,6,10)(3,11,14,15)(4,5,9,16)(7,13,8,12);(1,3)(2,7)(4,12)(5,11)(6,14)(8,10)(9,13)(15,16);(1,3)(2,8)(4,15)(5,13)(6,14)(7,10)(9,11)(12,16)Polycyclic presentation:
2^8|a^2=h,f^2=h,b^a=bd,c^a=ce,f^a=fg,e^b=efgh,g^b=gh,d^c=df,f^c=fh,g^c=gh,e^d=egh,f^d=fh,f^e=fh25 conjugacy classes — view the full character table.
| maximal subgroup of | #153614, #153615, #153617, … |
| maximal normal subgroup of | #153614, #153615, #153617, … |
| maximal quotient of | #151144, #151160, #151190, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6663);