Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
367 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 367 conjugacy classes of subgroups — too many to draw. Only its 119 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 16 points:
(5,6,7,8)(10,15,16,11);(1,2)(3,4)(9,10,13,15,14,16,12,11);(2,4)(9,11,12,10,14,15,13,16)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=d,d^2=h,e^2=h,b^a=bd,c^a=ce,c^b=cf,d^b=dh,f^b=fh,g^b=gh,e^c=eh,f^c=fh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #93951, #136152, #136153, … |
| maximal normal subgroup of | #93951, #136152, #136153, … |
| maximal quotient of | #120626, #120627, #120628, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4284);