Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13371 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #476 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C23 | AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(6,11,13,20)(7,15)(8,9)(12,16,14,19)(17,18);(1,2)(3,4)(6,12,13,14)(11,19,20,16);(2,4)(5,6,9,12,10,13,8,14)(7,16,15,11,18,19,17,20)Polycyclic presentation:
2^8|a^2=d,b^2=d,c^2=dg,e^2=g,f^2=h,g^2=h,b^a=bd,c^a=ce,e^a=eg,f^a=fh,g^a=gh,c^b=cf,d^c=dh,e^c=eg,f^c=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #125947, #125948, #125949, … |
| maximal normal subgroup of | #125947, #125948, #125949, … |
| maximal quotient of | #145796, #145826, #145862, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6445);