Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13697 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C4 |
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 | |
92 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)(3,6)(4,5)(7,8)(9,10)(11,23)(12,19)(13,18)(14,34)(15,36)(16,22)(17,21)(20,28)(24,37)(25,31)(26,33)(27,32)(29,39)(30,38)(35,40);(2,5)(6,8)(9,10,13,19,14,20,31,27,17,36,23,39,37,40,30,26,16,22,33,38,35,24,29,11,15,21,32,25,28,34,12,18);(1,3,4,7)(2,6,5,8)(9,11,12,24,28,38,32,22,15,26,29,40,35,39,33,36,16,27,30,20,37,19,23,10,17,18,31,34,14,25,13,21)Polycyclic presentation:
2^8|b^2=e,c^2=d,d^2=fg,e^2=f,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=df,e^a=efg,f^a=fg,g^a=gh,c^b=ch46 conjugacy classes — view the full character table.
| maximal subgroup of | #151574, #151575, #151576, … |
| maximal normal subgroup of | #151574, #151575, #151576, … |
| maximal quotient of | #152386, #152832, #152833, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6710);