Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1404 |
| automorphism | Aut(G) = #72286229 (order 2048) · Inn(G) = AΣL(1,𝔽16) · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | AΣL(1,𝔽16) |
| commutator | C2×Q8 | C22×C4 |
| Frattini, cosocle | #124 | C23 |
| socle | C2 | C2×AΣL(1,𝔽16) |
| prime radical | #3018 | C2 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
249 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 249 conjugacy classes of subgroups — too many to draw. Only its 45 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:
(1,2,4,8)(3,6,11,13)(5,10,15,16)(7,12,9,14);(1,2)(3,6)(4,13)(5,10)(7,15)(8,14)(9,12)(11,16);(1,3,5,12)(2,6,10,9)(4,11,15,14)(7,8,13,16)Polycyclic presentation:
2^8|a^2=e,c^2=h,d^2=h,f^2=h,b^a=bd,d^a=df,f^a=fg,g^a=gh,d^b=dh,e^b=ef,f^b=fh,g^b=gh,e^d=egh,f^d=fh,f^e=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #151915, #151916, #152027, … |
| maximal normal subgroup of | #151915, #151916, #152027, … |
| maximal quotient of | #149542, #149646, #149678, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6556);