Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13681 |
| automorphism | Aut(G) = C2×#512862 (order 1024) · Inn(G) = #2196 · Out(G) = C23 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2196 |
| commutator, Frattini, cosocle | C4⋊Q8 | C23 |
| prime radical | #3293 | C2 |
| LCS | G ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C4⋊Q8 ◁ G | |
| LECS | G ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
189 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 189 conjugacy classes of subgroups — too many to draw. Only its 29 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:
(1,2,8,7)(3,12,16,26)(4,17,19,21)(5,6,22,11)(9,20,15,14)(10,23,24,18)(13,25,29,31)(27,30,32,28);(1,3,8,16)(2,9,7,15)(4,13,19,29)(5,20,22,14)(6,12,11,26)(10,25,24,31)(17,30,21,28)(18,32,23,27);(1,4)(2,10)(3,13)(5,21)(6,23)(7,24)(8,19)(9,25)(11,18)(12,27)(14,30)(15,31)(16,29)(17,22)(20,28)(26,32)Polycyclic presentation:
2^8|a^2=h,b^2=h,d^2=h,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fg,d^b=dh,e^b=efgh,g^b=gh,d^c=df,e^c=eh,f^c=fh,g^c=gh,e^d=egh,f^d=fh,f^e=fh25 conjugacy classes — view the full character table.
| maximal subgroup of | #153620, #153621, #153623, … |
| maximal normal subgroup of | #153620, #153621, #153623, … |
| maximal quotient of | #151149, #151161, #151194, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6672);