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 | #2215 | C2 |
| 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 | |
75 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 40 points:
(1,2)(3,7,8,5)(4,6)(9,10,12,19,16,22,15,33)(11,23,26,39,29,17,28,34)(13,31,25,21,14,32,30,20)(18,37,35,24,38,40,36,27);(1,3)(2,5)(4,8)(6,7)(10,17)(11,24)(12,15)(13,30)(14,25)(18,21)(19,39)(20,38)(22,23)(26,37)(27,29)(28,40)(31,35)(32,36)(33,34)Polycyclic presentation:
2^8|a^2=d,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×#7420, #150248, #150249, … |
| maximal normal subgroup of | C2×#7420, #150248, #150249, … |
| maximal quotient of | #93786, #93809, #93848, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 408);