Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | Aut(G) = C25×#442 (order 2048) · Inn(G) = C2×D8 · Out(G) = C24×C4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D8 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C4×C8 | C23 |
| socle | C22 | C4×D8 |
| prime radical | C2×D8 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×C8 ◁ C82 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
113 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 113 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,3,5,6,8,7)(9,10,12,16)(11,14,20,22)(13,21,19,17)(15,24,23,18);(10,14)(11,18)(13,19)(15,17)(16,22)(20,24)(21,23);(1,3,4,6,5,7,8,2)Polycyclic presentation:
2^8|a^2=e,c^2=eh,d^2=fh,e^2=g,f^2=h,b^a=bd,c^a=cg,d^a=df,f^a=fh,d^b=df,f^b=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12070, #483778, #483784, … |
| maximal normal subgroup of | C2×#12070, #483778, #483784, … |
| maximal quotient of | #128704, #128745, #129186, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5058);