Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12173 |
| automorphism | Aut(G) = #72308726 (order 2048) · Inn(G) = #1598 · Out(G) = C24 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1598 |
| commutator, Frattini, cosocle | C2×M16 | C23 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
192 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 192 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) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,7)(4,8)(5,10)(6,11)(9,12)(13,16)(14,15);(2,7)(3,12)(4,5)(8,9)(10,15)(11,16)(13,14);(2,8,11,10)(7,9,16,15)Polycyclic presentation:
2^8|c^2=g,d^2=fh,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=df,e^b=egh,f^b=fh,d^c=dg,e^d=eh25 conjugacy classes — view the full character table.
| maximal subgroup of | #151166, #151168, #151252, … |
| maximal normal subgroup of | #151166, #151168, #151252, … |
| maximal quotient of | #123131, #123135, #123139, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5161);