Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | Aut(G) = C2×#493247 (order 1024) · Inn(G) = D8 · Out(G) = C22×C42 |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C16 | D8 |
| commutator | C8 | C2×C16 |
| Frattini, cosocle | C4×C16 | C22 |
| socle | C2 | #1370 |
| prime radical | C22 | #364 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16 ◁ C2×C16 ◁ C4×C16 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
46 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,5,9,4,12,17,26,7,11,22,25,19,28,18,31,8,13,23,27,20,29,16,32,6,24,21,30,15,10,3,14);(1,3,15,21,6,16,20,23,8,18,19,22,7,17,4,5)(2,9,12,26,11,25,28,31,13,27,29,32,24,30,10,14)Polycyclic presentation:
2^8|a^2=d,b^2=ce,c^2=eh,d^2=f,e^2=h,f^2=g,g^2=h,b^a=bc,c^a=ce,e^a=eh88 conjugacy classes — view the full character table.
| maximal subgroup of | #93751, #93752, #93866, … |
| maximal normal subgroup of | #93751, #93752, #93866, … |
| maximal quotient of | #93879, #93883, #94018, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 447);