Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13331 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #459 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
177 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 177 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:
(1,2,3,4)(5,6)(7,18)(8,15)(9,14)(10,19)(11,20)(12,17)(13,16);(5,7,9,17,11,19,10,20,12,14,18,6,13,15,8,16);(2,4)(6,14)(7,17)(8,9)(10,18)(11,13)(15,20)(16,19)Polycyclic presentation:
2^8|a^2=de,b^2=dg,d^2=gh,e^2=g,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fgh,g^a=gh,c^b=cf,f^b=fh,d^c=dg,e^c=egh,f^c=fg,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13478, #499337, #499338, … |
| maximal normal subgroup of | C2×#13478, #499337, #499338, … |
| maximal quotient of | #145900, #145902, #145911, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6466);