Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7418 |
| automorphism | Aut(G) = C2×#72225051 (order 2048) · Inn(G) = #357 · Out(G) = C22×C8 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #357 |
| commutator | C2×C16 | C2×C4 |
| Frattini, cosocle | C22×C16 | C22 |
| prime radical | C23 | SD32 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
70 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 72 points:
(1,2)(3,7,8,5)(4,6)(9,10,13,19,16,23,39,22)(11,24,26,58,29,61,37,60)(12,30,33,67,36,53,72,68)(14,40,38,20,15,42,32,21)(17,44,45,25,49,62,48,64)(18,50,52,35,55,70,54,34)(27,65,41,59,28,66,43,51)(31,56,69,46,63,57,71,47);(1,3)(2,5)(4,8)(6,7)(9,11,16,29)(10,17,23,49)(12,31,36,63)(13,37,39,26)(14,41,15,43)(18,51,55,59)(19,48,22,45)(20,56,21,57)(24,30,61,53)(25,34,64,35)(27,32,28,38)(33,71,72,69)(40,46,42,47)(44,50,62,70)(52,66,54,65)(58,68,60,67)Polycyclic presentation:
2^8|a^2=d,b^2=h,c^2=fg,d^2=h,e^2=gh,f^2=gh,g^2=h,b^a=bc,c^a=ce,e^a=eg,f^a=fg,g^a=gh,c^b=cf,d^b=defg,e^b=eg,f^b=fg,g^b=gh49 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7421, #150248, #150249, … |
| maximal normal subgroup of | C2×#7421, #150248, #150249, … |
| maximal quotient of | #93787, #93808, #93849, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 409);