Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2230 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #476 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C23 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
183 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 183 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,4)(5,6,8,14,10,18,9,13,12,15,19,20,11,7,16,17);(1,2,3,4)(5,7,11,6,12,18,10,15)(8,14,16,13,19,20,9,17);(5,8,10,9,12,19,11,16)(6,13,7,14,15,17,18,20)Polycyclic presentation:
2^8|a^2=f,b^2=def,c^2=e,d^2=g,e^2=h,f^2=gh,g^2=h,b^a=bd,c^a=ce,e^a=eh,d^b=dg,f^b=fgh,g^b=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #136857, #136859, #136861, … |
| maximal normal subgroup of | #136857, #136859, #136861, … |
| maximal quotient of | #141847, #141869, #142078, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6260);