Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
689 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,7,8)(3,4,5,6)(25,26)(27,31)(28,36)(29,30)(32,37)(33,34)(35,40)(38,39);(1,3,7,5)(2,4,8,6)(10,14)(12,20)(15,17)(18,24)(26,30)(27,33)(28,37)(38,40);(1,4,7,6)(2,3,8,5)(9,10,13,14)(11,17,19,15)(12,16,20,21)(18,23,24,22)(25,26)(27,34)(28,32)(29,30)(31,33)(35,40)(36,37)(38,39);(1,5,7,3)(2,4,8,6)(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,27,29,33)(26,31,30,34)(28,35,37,39)(32,38,36,40);(1,6,7,4)(2,3,8,5)(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,28)(26,32)(27,35)(29,37)(30,36)(31,38)(33,39)(34,40)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=fgh,d^2=gh,e^2=g,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df,e^b=egh,d^c=dgh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#58727, #8104867, #8104897, … |
| maximal normal subgroup of | C2×#58727, #8104867, #8104897, … |
| maximal quotient of | #586128, #587261, #587295, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51715);