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 | |
687 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,3)(4,7,8,6)(10,14)(11,17)(16,21)(18,23)(25,26,31,28)(27,29,32,35)(30,33,39,36)(34,37,38,40);(2,3)(6,7)(10,14)(15,20)(16,21)(22,24)(25,27,31,32)(26,29,28,35)(30,38,39,34)(33,40,36,37);(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24)(25,28,31,26)(27,29,32,35)(30,36,39,33)(34,37,38,40);(1,3,5,2)(4,7,8,6)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,29)(26,32)(27,28)(30,37)(31,35)(33,38)(34,36)(39,40);(1,4)(2,6)(3,7)(5,8)(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,30)(26,33)(27,34)(28,36)(29,37)(31,39)(32,38)(35,40)Polycyclic presentation:
2^8|a^2=fgh,b^2=gh,c^2=h,d^2=f,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×#57778, #8101754, #8101964, … |
| maximal normal subgroup of | C2×#57778, #8101754, #8101964, … |
| maximal quotient of | #586718, #586996, #587310, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 50766);