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 | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
705 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,7)(3,6,8,5)(9,10,13,14)(11,17,18,15)(12,20,21,16)(19,22,24,23)(25,26)(27,31)(28,35)(29,30)(32,36)(33,39)(34,37)(38,40);(1,3,4,8)(2,5,7,6)(10,14)(11,18)(16,20)(19,24)(26,30)(31,37)(32,35)(38,39);(3,8)(5,6)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,27,29,34)(26,31,30,37)(28,33,36,40)(32,38,35,39);(1,3)(2,6)(4,8)(5,7)(9,11,13,18)(10,15,14,17)(12,19,21,24)(16,22,20,23)(25,26)(27,31)(28,32)(29,30)(33,38)(34,37)(35,36)(39,40);(1,2)(3,6)(4,7)(5,8)(9,12,13,21)(10,16,14,20)(11,19,18,24)(15,22,17,23)(25,28)(26,32)(27,33)(29,36)(30,35)(31,38)(34,40)(37,39)Polycyclic presentation:
2^8|a^2=h,b^2=gh,c^2=fh,d^2=g,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×#57886, #8101727, #8101959, … |
| maximal normal subgroup of | C2×#57886, #8101727, #8101959, … |
| maximal quotient of | #586114, #586632, #587073, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 50874);