Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1357 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #331 |
| commutator | C42 | C42 |
| Frattini, cosocle | C2×C4×C8 | C22 |
| prime radical | C23 | #100 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
119 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2,3,4)(6,10)(7,14)(8,15)(11,18)(12,13)(16,20);(5,6,7,11,15,13,20,19,9,12,16,18,8,10,14,17)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=f,e^2=fgh,g^2=h,b^a=bc,c^a=cf,e^a=efg,g^a=gh,c^b=cg,f^b=fh,g^b=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #94682, #94683, #94730, … |
| maximal normal subgroup of | #94682, #94683, #94730, … |
| maximal quotient of | #93064, #93065, #93068, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 257);