Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C23×C4 | C23 |
| commutator | C22 | C2×C4×C8 |
| socle, prime radical | C24 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
487 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6)(7,8)(13,14,17,15,16,18,20,19);(2,4)(9,10,11,12);(6,8)(13,15,17,18,16,19,20,14)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=f,f^2=h,b^a=bd,c^a=ce112 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10876, #353862, #353864, … |
| maximal normal subgroup of | C2×#10876, #353862, #353864, … |
| maximal quotient of | #109460, #109462, #109470, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3864);