Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1404 |
| automorphism | Aut(G) = C4×#33553 (order 1024) · Inn(G) = AΣL(1,𝔽16) · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | AΣL(1,𝔽16) |
| commutator | C2×Q8 | C2×C8 |
| Frattini, cosocle | #408 | C22 |
| socle | C2 | #1326 |
| prime radical | C2×#41 | C2×C4 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #408 ◁ G | |
| LECS | G ▷ #408 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
85 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,5,11,7,12,26,42,8,13,27,43,6,28,30,41)(3,14,16,44,18,46,24,57,19,36,51,60,17,45,53,63)(4,20,21,54,22,10,35,37,23,38,25,58,29,39,56,64)(9,31,32,50,33,59,61,52,34,55,62,15,47,48,40,49);(1,3,6,17,8,19,7,18)(2,9,12,33,13,34,28,47)(4,15,22,49,23,50,29,52)(5,24,26,51,27,53,30,16)(10,36,20,46,39,14,38,45)(11,40,41,62,43,61,42,32)(21,55,56,59,25,31,35,48)(37,63,58,44,64,57,54,60)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=h,d^2=f,e^2=h,f^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gh,c^b=cgh,d^b=de,g^b=gh,d^c=dgh,e^c=eh,e^d=eh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7378, #149652, #149683, … |
| maximal normal subgroup of | C2×#7378, #149652, #149683, … |
| maximal quotient of | #93733, #93743, #93952, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 366);