Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1402 |
| automorphism | Aut(G) = #72286229 (order 2048) · Inn(G) = AΣL(1,𝔽16) · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | AΣL(1,𝔽16) |
| commutator | C2×D4 | C42 |
| Frattini, prime radical, cosocle | #532 | C22 |
| socle | C2 | #1390 |
| LCS | G ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
143 conjugacy classes of subgroups; only the 33 normal subgroups are shown.
This group has 143 conjugacy classes of subgroups — too many to draw. Only its 33 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 — 2 generators on 16 points:
(1,2,4,8)(3,12,13,10)(5,9,15,16)(6,11,14,7);(1,2,5,9,6,11,13,10)(3,7,4,12,15,8,14,16)Polycyclic presentation:
2^8|a^2=d,b^2=e,e^2=h,f^2=h,b^a=bc,c^a=cf,f^a=fg,g^a=gh,d^b=dfh,f^b=fh,g^b=gh,d^c=dgh,f^c=fh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7486, #151794, #151828, … |
| maximal normal subgroup of | C2×#7486, #151794, #151828, … |
| maximal quotient of | #94087, #94131, #94143, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 474);