Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7418 |
| automorphism | Aut(G) = C2×D4×#12067 (order 4096) · Inn(G) = #357 · Out(G) = C23×C8 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22 | #357 |
| commutator | C2×C16 | C2×C4 |
| Frattini, cosocle | C4×C16 | C22 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ C4×C16 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
66 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 40 points:
(1,2,3,6,4,5,7,8)(10,15)(11,19)(12,21)(13,25)(16,31)(17,34)(18,30)(20,24)(22,38)(23,28)(26,32)(27,29)(33,39)(35,36)(37,40);(2,5)(3,7)(9,10,14,18)(11,20,23,37)(12,24,26,40)(13,27,25,17)(15,28,30,19)(16,32,33,21)(22,39,35,31)(29,36,34,38)Polycyclic presentation:
2^8|a^2=d,b^2=h,c^2=fg,d^2=efg,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×#7445, #150938, #150939, … |
| maximal normal subgroup of | C2×#7445, #150938, #150939, … |
| maximal quotient of | #93863, #93870, #93873, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 433);