Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13697 |
| automorphism | Aut(G) order 16384 |
| 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 | #2246 | C2 |
| 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 | |
97 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)(10,11)(12,13)(14,25)(15,30)(17,27)(18,22)(19,35)(20,34)(21,24)(23,36)(26,33)(28,29)(31,39)(32,38)(37,40);(2,5)(6,8)(9,10,17,32,14,19,26,23,30,36,29,35,39,38,12,11,16,21,13,18,31,37,28,34,15,20,33,40,25,22,27,24);(1,3,4,7)(2,6,5,8)(9,11,12,22,25,35,29,20,15,23,26,37,31,32,17,21,16,24,27,38,39,40,33,36,30,34,28,19,14,18,13,10)Polycyclic presentation:
2^8|b^2=eh,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 | #151550, #151551, #151562, … |
| maximal normal subgroup of | #151550, #151551, #151562, … |
| maximal quotient of | #152386, #152832, #152892, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6711);