Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×D4 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C25 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
759 conjugacy classes of subgroups; only the 223 normal subgroups are shown.
This group has 759 conjugacy classes of subgroups — too many to draw. Only its 223 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) = 128 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6)(7,8)(9,10,11,12)(14,18)(15,16)(19,20);(2,4)(13,14,16,20,17,19,15,18);(6,8)(14,19)(18,20)Polycyclic presentation:
2^8|a^2=f,b^2=g,d^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=gh,c^b=ch64 conjugacy classes — view the full character table.
| maximal subgroup of | #93537, #93538, #105894, … |
| maximal normal subgroup of | #93537, #93538, #105894, … |
| maximal quotient of | #105498, #105500, #105506, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3710);