Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12307 |
| automorphism | Aut(G) = #72308726 (order 2048) · Inn(G) = #1616 · Out(G) = C24 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1616 |
| commutator, Frattini, cosocle | C2×M16 | C23 |
| prime radical | C23 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
137 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 137 conjugacy classes of subgroups — too many to draw. Only its 35 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:
(1,2,5,8,6,11,22,10)(3,12,14,27,15,30,21,29)(4,16,19,9,20,26,18,28)(7,17,24,13,25,31,23,32);(1,3,6,15)(2,7,11,25)(4,17,20,31)(5,21,22,14)(8,23,10,24)(9,27,28,29)(12,16,30,26)(13,19,32,18);(2,8,11,10)(7,23,25,24)(9,16,28,26)(12,27,30,29)Polycyclic presentation:
2^8|a^2=fg,b^2=h,c^2=g,d^2=f,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fh,g^a=gh,d^b=dfh,e^b=egh,f^b=fh,d^c=dg,e^d=eh25 conjugacy classes — view the full character table.
| maximal subgroup of | #151217, #151219, #151253, … |
| maximal normal subgroup of | #151217, #151219, #151253, … |
| maximal quotient of | #126352, #126392, #126449, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5296);