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) = C2 |
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 | |
134 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10,15,16,13,18,12,14)(11,19,22,23,21,17,20,24);(1,3,4,7)(2,5,6,8)(10,16)(11,19,21,17)(12,15)(14,18)(20,23,22,24);(2,6)(5,8)(9,11,12,20,13,21,15,22)(10,17,14,23,18,19,16,24)Polycyclic presentation:
2^8|a^2=dh,b^2=deg,c^2=d,d^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,c^b=cf,d^b=dh,f^b=fg,g^b=gh,f^c=fg,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #126485, #126486, #126487, … |
| maximal normal subgroup of | #126485, #126486, #126487, … |
| maximal quotient of | #146287, #148719, #148979, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6531);