Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62307 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | #3543 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #529 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
887 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,10)(4,8)(5,14)(6,9)(7,12)(11,16)(13,15)(19,25)(20,27)(22,28)(24,30)(26,31)(29,32);(1,3)(2,7)(4,11)(5,13)(6,12)(8,15)(9,10)(14,16)(17,18)(19,22)(20,24)(21,23)(25,28)(26,29)(27,30)(31,32);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,19,20,25,21,26,27,31)(18,22,24,28,23,29,30,32);(3,12)(7,10)(11,13)(15,16)(19,26)(22,29)(25,31)(28,32);(1,5,6,4)(2,8,9,14)(3,13,12,11)(7,15,10,16)(18,23)(19,26)(24,30)(25,31)Polycyclic presentation:
2^8|c^2=fg,e^2=f,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,d^b=df,e^b=eh,d^c=dh,e^c=eh49 conjugacy classes — view the full character table.
| maximal subgroup of | #6367616, #6367619, #6367625, … |
| maximal normal subgroup of | #6367616, #6367619, #6367625, … |
| maximal quotient of | #6386968, #6389674, #6389691, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55596);