Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2212 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #459 |
| 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 | |
155 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 155 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) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,4,3)(6,12)(7,8)(9,10)(13,20)(14,19)(15,16)(17,18);(1,3,4,2)(5,6)(7,12)(8,17)(9,13)(10,18)(11,14)(15,20)(16,19);(5,7,10,15,11,16,9,8)(6,12,18,20,14,19,13,17)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=eh,d^2=gh,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,d^b=dg,g^b=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #136324, #136325, #136327, … |
| maximal normal subgroup of | #136324, #136325, #136327, … |
| maximal quotient of | #140381, #140402, #140419, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6169);