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 | |
1456 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2)(3,4)(5,6,9,12)(7,13,15,19)(8,17,18,11)(10,14,20,16);(5,7)(6,10)(8,14)(9,15)(11,19)(12,20)(13,17)(16,18);(2,4);(7,14)(8,18)(10,19)(11,17)(13,20)(15,16);(7,15)(10,20)(13,19)(14,16)Polycyclic presentation:
2^8|a^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 | #6533064, #6533067, #6533076, … |
| maximal normal subgroup of | #6533064, #6533067, #6533076, … |
| maximal quotient of | #6358723, #6358729, #6358730, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54054);