Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
770 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(9,10,13,16)(11,14,12,15)(17,18)(19,26)(20,21)(22,23)(24,27)(25,31)(28,29)(30,32);(2,5)(3,7)(9,11,13,12)(10,14,16,15)(18,23)(19,27)(21,31)(28,30);(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,15)(13,16)(17,19)(18,24)(20,30)(21,29)(22,27)(23,26)(25,28)(31,32);(1,2)(3,6)(4,5)(7,8)(17,20,22,25)(18,21,23,31)(19,28,27,30)(24,32,26,29);(1,3)(2,6)(4,7)(5,8)(9,12,13,11)(10,15,16,14)(17,21)(18,25)(19,29)(20,23)(22,31)(24,28)(26,30)(27,32)Polycyclic presentation:
2^8|a^2=fg,b^2=fg,c^2=fh,d^2=f,e^2=fg,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#49908, #7705661, #7705679, … |
| maximal normal subgroup of | C2×#49908, #7705661, #7705679, … |
| maximal quotient of | #560731, #560942, #561816, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 42896);