Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13694 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22 | C2×D16 |
| commutator, Frattini, cosocle | C2×C16 | C23 |
| 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 | |
91 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,17,11)(12,28,30,34)(13,25,33,21)(14,24,18,20)(15,35,36,40)(16,27,19,23)(22,38,26,31)(29,32,39,37);(2,5)(6,8)(9,11,14,25,15,26,32,30,19,23,28,39,38,35,13,24,17,10,18,21,36,22,37,12,16,27,34,29,31,40,33,20);(1,3,4,7)(2,6,5,8)(9,12,13,26,31,21,28,11,16,24,32,40,36,39,14,27,17,30,33,22,38,25,34,10,19,20,37,35,15,29,18,23)Polycyclic presentation:
2^8|a^2=h,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=gh70 conjugacy classes — view the full character table.
| maximal subgroup of | #151546, #151547, #151548, … |
| maximal normal subgroup of | #151546, #151547, #151548, … |
| maximal quotient of | #152384, #152831, #152889, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6708);