Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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 | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
758 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(11,15)(14,16)(17,18,21,22)(19,25,27,23)(20,24,28,29)(26,31,32,30);(1,2,4,6)(3,5,8,7)(9,10,13,12)(11,14,15,16)(18,22)(23,25)(24,29)(30,31);(3,8)(5,7)(9,11)(10,14)(12,16)(13,15)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32);(1,3,4,8)(2,5,6,7)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31);(1,2)(3,5)(4,6)(7,8)(9,12,13,10)(11,16,15,14)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)Polycyclic presentation:
2^8|a^2=h,b^2=fg,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#41654, #7639826, #7640239, … |
| maximal normal subgroup of | C2×#41654, #7639826, #7640239, … |
| maximal quotient of | #544787, #545239, #545865, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 34642);