Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1402 |
| 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×D4 | C2×C8 |
| Frattini, cosocle | #408 | C22 |
| socle | C2 | #1326 |
| prime radical | C2×#41 | C2×C4 |
| LCS | G ▷ C2×D4 ▷ 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:
(1,2,5,11,7,12,27,42,8,13,28,43,6,29,31,41)(3,14,16,44,18,46,50,64,19,47,51,60,17,45,25,57)(4,20,22,53,23,10,36,37,24,38,26,58,30,39,56,63)(9,32,33,15,34,48,40,49,35,59,62,21,54,55,61,52);(1,3,6,17,8,19,7,18)(2,9,12,34,13,35,29,54)(4,21,23,52,24,15,30,49)(5,25,27,16,28,50,31,51)(10,14,20,45,39,47,38,46)(11,40,41,33,43,61,42,62)(22,32,56,55,26,59,36,48)(37,57,58,44,63,64,53,60)Polycyclic presentation:
2^8|a^2=d,b^2=e,d^2=f,e^2=h,f^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gh,c^b=cg,d^b=deh,g^b=gh,d^c=dgh,e^c=eh,e^d=eh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7377, #149650, #149681, … |
| maximal normal subgroup of | C2×#7377, #149650, #149681, … |
| maximal quotient of | #93732, #93744, #93952, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 365);