Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2180 |
| automorphism | Aut(G) = #495843 (order 512) · Inn(G) = #442 · Out(G) = D4 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #442 |
| commutator | C8 | C22×C8 |
| Frattini, cosocle | C4×C8 | C23 |
| socle | C2 | #1580 |
| prime radical | #508 | C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ #346 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
102 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,11,7,12,17,23,8,13,5,16,22,3,14,10)(4,15,18,29,19,9,24,26,20,27,21,25,32,28,30,31);(1,3)(2,7)(4,15)(5,10)(6,23)(8,12)(9,19)(11,14)(13,22)(16,17)(18,25)(20,27)(21,29)(24,31)(26,30)(28,32);(1,4,17,24,22,32,6,18,8,20,14,30,7,19,5,21)(2,9,10,29,3,15,16,31,13,28,23,25,12,27,11,26)Polycyclic presentation:
2^8|a^2=e,c^2=ef,d^2=f,e^2=g,f^2=h,g^2=h,b^a=bd,c^a=cf,d^a=df,f^a=fh,d^b=dfh,e^b=eh,f^b=fh,d^c=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | #151105, AGL(1,ℤ/32), #151119, … |
| maximal normal subgroup of | #151105, AGL(1,ℤ/32), #151119, … |
| maximal quotient of | #122961, #122962, #122963, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5056);