Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7523 |
| automorphism | Aut(G) = C2×#13569 (order 512) · Inn(G) = #1404 · Out(G) = C22 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1404 |
| commutator | #125 | C2×C4 |
| Frattini, cosocle | #491 | C22 |
| prime radical | C22 | AΣL(1,𝔽16) |
| LCS | G ▷ #125 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #491 ◁ G | |
| LECS | G ▷ #491 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
63 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,5,10,6,12,3,13)(4,16,18,26,20,27,32,8)(7,11,15,30,22,31,21,29)(9,17,25,23,28,14,24,19);(1,3,6,5)(2,8,12,26)(4,17,20,14)(7,21,22,15)(9,13,28,10)(11,25,31,24)(16,29,27,30)(18,19,32,23)Polycyclic presentation:
2^8|a^2=d,b^2=h,c^2=g,d^2=h,e^2=gh,f^2=h,b^a=bc,c^a=ce,e^a=ef,f^a=fg,g^a=gh,c^b=cg,d^b=deh,e^b=egh,f^b=fg,d^c=dfg,e^c=eg,f^c=fh,e^d=egh,f^d=fh19 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7526, #153489, #153490, … |
| maximal normal subgroup of | C2×#7526, #153489, #153490, … |
| maximal quotient of | #94545, #94600, #94665, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 514);