Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2180 |
| automorphism | Aut(G) = #72231459 (order 2048) · Inn(G) = #442 · Out(G) = #120 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #442 |
| commutator | C8 | C22×C8 |
| Frattini, cosocle | C4×C8 | C23 |
| prime radical | C2×D16 | C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #322 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
123 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,7,5,3,6,8)(9,10,12,15,13,18,23,17)(11,19,22,16,21,14,20,24);(10,14)(11,20)(12,23)(15,16)(17,24)(18,19)(21,22);(1,3,4,8,5,2,6,7)(10,15,18,17)(11,21)(14,16,19,24)(20,22)Polycyclic presentation:
2^8|a^2=e,c^2=efh,d^2=f,e^2=g,f^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 | AGL(1,ℤ/32), C2×#12067, #483589, … |
| maximal normal subgroup of | AGL(1,ℤ/32), C2×#12067, #483589, … |
| maximal quotient of | #122963, #122966, #122968, … |
| automorphism group of | SD64 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5055);