Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | Aut(G) order 2048 |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | D8 |
| commutator | C8 | C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C22 |
| socle | C22 | D4⋊C8 |
| prime radical | C23 | #104 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
137 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2)(3,4)(5,6,8,14,10,15,17,13,11,16,7,12,9,19,18,20);(2,4)(6,12,15,20,16,14,19,13)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=fh,d^2=g,e^2=fg,f^2=h,g^2=h,b^a=bc,c^a=cf,e^a=efh,f^a=fh88 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7179, #126789, #126850, … |
| maximal normal subgroup of | C2×#7179, #126789, #126850, … |
| maximal quotient of | #92931, #92932, #92937, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 167);