Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13569 |
| automorphism | Aut(G) = #72236721 (order 2048) · Inn(G) = C2×AΣL(1,𝔽16) · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×AΣL(1,𝔽16) |
| commutator | C2×Q8 | C22×C4 |
| Frattini, cosocle | C4⋊Q8 | C23 |
| prime radical | #3018 | C2 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
243 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 243 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:
(1,2,5,10,7,3,12,14)(4,8,15,13,6,11,16,9);(1,3)(2,7)(4,8)(5,9)(6,11)(10,16)(12,13)(14,15);(1,4,7,6)(2,8,3,11)(5,15,12,16)(9,10,13,14)Polycyclic presentation:
2^8|a^2=e,c^2=h,d^2=h,e^2=h,f^2=h,b^a=bd,d^a=df,f^a=fg,g^a=gh,c^b=ch,d^b=dh,e^b=ef,f^b=fh,g^b=gh,e^d=egh,f^d=fh,f^e=fh28 conjugacy classes — view the full character table.
| maximal subgroup of | #152027, #152028, #152180, … |
| maximal normal subgroup of | #152027, #152028, #152180, … |
| maximal quotient of | #149544, #149648, #149679, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6562);