Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12087 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #442 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
175 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 175 conjugacy classes of subgroups — too many to draw. Only its 59 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)(3,4)(5,6,11,15,12,16,19,7)(8,10,18,13,14,9,17,20);(5,7,9,18,11,16,20,8,12,15,10,17,19,6,13,14);(2,4)(5,8,9,7,11,17,20,16,12,14,10,15,19,18,13,6)Polycyclic presentation:
2^8|a^2=f,b^2=dfgh,c^2=d,d^2=g,e^2=g,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gh,c^b=cg,d^b=dh,e^b=eh,f^b=fh,f^c=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13155, #494928, #494929, … |
| maximal normal subgroup of | C2×#13155, #494928, #494929, … |
| maximal quotient of | #140188, #140210, #140283, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6143);