Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1044 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 16 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,13)(12,14)(15,16);(1,3)(2,6)(4,7)(5,8);(1,4,5,2)(3,7,8,6)(10,13)(11,16)(14,15);(3,8)(6,7);(10,14)(13,15)Polycyclic presentation:
2^8|a^2=f,c^2=f,g^2=h,b^a=bf,c^a=cg,e^a=eh,g^a=gh,d^b=df,g^c=gh55 conjugacy classes — view the full character table.
| maximal subgroup of | #250923, #250955, #250987, … |
| maximal normal subgroup of | #250923, #250955, #250987, … |
| maximal quotient of | #6379292, #6379298, #6379869, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54240);