Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13371 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #476 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
201 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 201 conjugacy classes of subgroups — too many to draw. Only its 57 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:
(6,11,13,19)(7,15)(8,9)(12,20,14,16)(17,18);(1,2)(3,4)(6,12,13,14)(11,16,19,20);(2,4)(5,6)(7,16)(8,14)(9,12)(10,13)(11,17)(15,19)(18,20)Polycyclic presentation:
2^8|a^2=d,b^2=d,e^2=g,f^2=h,g^2=h,b^a=bd,c^a=ce,e^a=eg,f^a=fh,g^a=gh,c^b=cf,d^c=dh,e^c=egh,f^c=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #125892, #125893, #125894, … |
| maximal normal subgroup of | #125892, #125893, #125894, … |
| maximal quotient of | #144452, #144482, #144521, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6359);