Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | Aut(G) = C2×C4×AGL(1,ℤ/16) (order 1024) · Inn(G) = D8 · Out(G) = C22×C42 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C16 | D8 |
| commutator | C8 | C22×C8 |
| Frattini, cosocle | C4×C8 | C23 |
| socle | C2 | C8×D8 |
| prime radical | #508 | C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16 ◁ C2×C16 ◁ C4×C16 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,12,8,13,24,11,9,14,5,23,7,3,15,17)(4,10,19,26,21,27,32,25,22,28,18,31,20,16,29,30);(1,3)(2,8)(4,16)(5,11)(6,17)(7,14)(9,13)(10,21)(12,24)(15,23)(18,25)(19,30)(20,28)(22,27)(26,32)(29,31);(1,4,6,19,8,21,24,32,9,22,5,18,7,20,15,29)(2,10,12,26,13,27,11,25,14,28,23,31,3,16,17,30)Polycyclic presentation:
2^8|a^2=e,c^2=e,d^2=fh,e^2=g,f^2=h,g^2=h,b^a=bd,d^a=df,f^a=fh,d^b=df,f^b=fh88 conjugacy classes — view the full character table.
| maximal subgroup of | #93751, #93752, #151072, … |
| maximal normal subgroup of | #93751, #93752, #151072, … |
| maximal quotient of | C16×D16, C16×SD32, C16×Q32, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5034);