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 | |
1415 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)(6,11)(7,9)(12,19)(13,17)(14,18)(15,20);(5,6)(7,11)(8,15)(9,17)(10,13)(12,14)(16,19)(18,20);(2,4)(5,7,10,9)(6,11,13,17)(8,14,16,18)(12,19,20,15);(5,8,10,16)(6,12,13,20)(7,14,9,18)(11,19,17,15);(6,13)(11,17)(12,20)(15,19)Polycyclic presentation:
2^8|c^2=h,d^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 | #250913, #250945, #250977, … |
| maximal normal subgroup of | #250913, #250945, #250977, … |
| maximal quotient of | #6358727, #6358734, #6358741, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54058);