Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2199 |
| automorphism | Aut(G) = C22×#60399 (order 1024) · Inn(G) = B(4,𝔽2) · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | B(4,𝔽2) |
| commutator | C2×D4 | C22×C4 |
| Frattini, cosocle | C2×#41 | C23 |
| socle | C2 | #1889 |
| prime radical | #3025 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
322 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 322 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) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,9,6,10,17,23)(3,11,15,25,13,24,28,32)(4,8,12,19,16,22,26,30)(7,18,21,27,20,29,31,14);(1,3,12,18,6,13,26,29)(2,7,19,25,10,20,30,32)(4,14,5,15,16,27,17,28)(8,11,9,21,22,24,23,31);(3,13)(4,16)(5,17)(7,20)(11,24)(18,29)(21,31)(25,32)Polycyclic presentation:
2^8|a^2=f,b^2=df,f^2=h,b^a=bd,c^a=ce,e^a=eg,g^a=gh,c^b=ch,e^b=egh,d^c=dg,f^c=fg,e^d=eh,f^e=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13046, #494587, #494588, … |
| maximal normal subgroup of | C2×#13046, #494587, #494588, … |
| maximal quotient of | #136819, #136820, #136845, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6034);