Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13657 |
| automorphism | Aut(G) = C2×#483875 (order 1024) · Inn(G) = #2177 · Out(G) = C23 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2177 |
| commutator | C16 | C22×C4 |
| Frattini, cosocle | C2×C16 | C23 |
| prime radical | #2263 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #346 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
97 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,8,10,24)(3,13)(4,16)(5,20)(6,14)(9,17,21,25)(11,23,28,27)(12,29,22,26)(15,19)(18,30)(31,32);(1,2)(3,9)(4,17)(5,21)(6,22)(7,10)(8,15)(11,20)(12,14)(13,28)(16,24)(18,23)(19,25)(26,32)(27,31)(29,30);(1,3,14,20,7,5,6,13)(2,9,12,11,10,21,22,28)(4,16,31,30,19,15,18,32)(8,23,26,17,24,27,29,25)Polycyclic presentation:
2^8|a^2=e,c^2=egh,d^2=fgh,f^2=gh,g^2=h,b^a=bd,c^a=cg,d^a=df,f^a=fg,g^a=gh,d^b=dfh,e^b=eh,f^b=fg,g^b=gh,d^c=dh34 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,ℤ/32), #151119, C2×#13658, … |
| maximal normal subgroup of | AGL(1,ℤ/32), #151119, C2×#13658, … |
| maximal quotient of | #151083, #151085, #151115, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6646);