Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13371 |
| automorphism | Aut(G) = C24×#3109 (order 2048) · Inn(G) = #476 · Out(G) = C25 |
| 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 | |
135 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 135 conjugacy classes of subgroups — too many to draw. Only its 55 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:
(5,6,11,12,18,8,7,13,10,15,19,17,9,14,16,20);(1,2)(3,4)(5,7,9,11,10,16,18,19)(6,12,8,13,15,17,14,20);(2,4)(5,8,9,15,10,14,18,6)(7,13,11,17,16,20,19,12)Polycyclic presentation:
2^8|a^2=eg,b^2=d,c^2=d,d^2=h,e^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,e^c=eg,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13521, #499439, #499440, … |
| maximal normal subgroup of | C2×#13521, #499439, #499440, … |
| maximal quotient of | #146166, #146169, #146181, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6509);