Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13661 |
| automorphism | Aut(G) = C24×#12067 (order 4096) · Inn(G) = C2×D16 · Out(G) = C23×C8 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D16 |
| commutator, Frattini, cosocle | C2×C16 | C23 |
| prime radical | C2×D16 | C22 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ C4×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
85 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10,11,15)(12,16,20,28)(13,26,21,38)(14,27,22,19)(17,33,29,23)(18,35,30,24)(25,36,37,32)(31,40,34,39);(1,3,4,8)(2,5,6,7)(10,15)(16,28)(17,29)(18,30)(19,27)(26,38)(31,34)(32,36);(10,16)(12,23)(13,24)(14,22)(15,28)(17,18)(19,36)(20,33)(21,35)(25,40)(26,31)(27,32)(29,30)(34,38)(37,39)Polycyclic presentation:
2^8|a^2=d,b^2=dh,e^2=fg,f^2=gh,g^2=h,b^a=bd,c^a=ce,e^a=ef,f^a=fg,g^a=gh,e^c=ef,f^c=fg,g^c=gh49 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13707, #512960, #512961, … |
| maximal normal subgroup of | C2×#13707, #512960, #512961, … |
| maximal quotient of | #152271, #152433, #152435, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6695);