Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13357 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×Q8 | C23 |
| prime radical | C23 | #120 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
222 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 222 conjugacy classes of subgroups — too many to draw. Only its 45 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:
(2,3)(5,6,9,11)(7,14,13,10)(8,15,17,20)(12,18,19,16);(1,2)(3,4)(5,7,9,13)(6,10,11,14)(8,16,17,18)(12,15,19,20);(1,3,4,2)(5,8,9,17)(6,12,11,19)(7,15,13,20)(10,18,14,16)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=d,e^2=h,f^2=h,b^a=bd,c^a=ce,e^a=eh,f^a=fgh,c^b=cf,e^b=eg,f^b=fh,e^c=eh,f^c=fh,g^c=gh,f^e=fh31 conjugacy classes — view the full character table.
| maximal subgroup of | #151226, #151227, #151234, … |
| maximal normal subgroup of | #151226, #151227, #151234, … |
| maximal quotient of | #144669, #144705, #144736, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6416);