Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7408 |
| automorphism | Aut(G) = C23×#2196 (order 1024) · Inn(G) = AΣL(1,𝔽16) · Out(G) = C24 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | AΣL(1,𝔽16) |
| commutator | C2×C4⋊C4 | C2×C4 |
| Frattini, cosocle | #391 | C22 |
| prime radical | C23 | #99 |
| LCS | G ▷ C2×C4⋊C4 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #391 ◁ G | |
| LECS | G ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
103 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 24 points:
(2,5,6,7)(9,10,13,17,15,11,19,20)(12,22,14,18,23,24,16,21);(1,2)(3,7)(4,6)(5,8)(9,11,14,21,15,10,16,18)(12,20,19,24,23,17,13,22)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=fg,d^2=h,e^2=h,b^a=bc,c^a=ce,e^a=eg,f^a=fh,g^a=gh,c^b=cf,d^b=defgh,f^b=fh,g^b=gh,d^c=dgh,e^c=eh,e^d=eh31 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7431, #150275, #150276, … |
| maximal normal subgroup of | C2×#7431, #150275, #150276, … |
| maximal quotient of | #93238, #93239, #93951, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 419);