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) = C29 |
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 | |
1389 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2)(3,4)(5,6,10,8)(7,13,16,20)(9,18,19,12)(11,15,14,17);(5,7)(6,11)(8,14)(9,15)(10,16)(12,20)(13,18)(17,19);(2,4)(5,8,10,6)(7,14,16,11)(9,18,19,12)(13,17,20,15);(7,15)(9,19)(11,20)(12,18)(13,14)(16,17);(7,16)(11,14)(13,20)(15,17)Polycyclic presentation:
2^8|a^2=h,c^2=h,f^2=h,b^a=bf,c^a=cg,f^a=fh,d^b=df,e^b=eh,f^b=fh,f^d=fh55 conjugacy classes — view the full character table.
| maximal subgroup of | #250905, #250937, #250969, … |
| maximal normal subgroup of | #250905, #250937, #250969, … |
| maximal quotient of | #6358724, #6358731, #6358738, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54055);