Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | Aut(G) order 65536 |
| 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 | C25 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
726 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,8)(4,7)(5,6)(10,14)(11,17)(16,21)(19,23)(25,26)(27,34)(28,31)(29,37)(30,33)(32,38)(35,39)(36,40);(1,3)(2,7)(4,6)(5,8)(12,20)(16,21)(19,23)(22,24)(25,27,30,28)(26,31,33,34)(29,35,38,36)(32,39,37,40);(3,4)(7,8)(9,10,13,14)(11,18,17,15)(12,16,20,21)(19,24,23,22)(25,28,30,27)(26,31,33,34)(29,36,38,35)(32,39,37,40);(1,4,6,3)(2,7,5,8)(9,11,13,17)(10,15,14,18)(12,19,20,23)(16,22,21,24)(25,26)(27,34)(28,31)(29,32)(30,33)(35,40)(36,39)(37,38);(1,5,6,2)(3,7,4,8)(9,12,13,20)(10,16,14,21)(11,19,17,23)(15,22,18,24)(25,29,30,38)(26,32,33,37)(27,35,28,36)(31,39,34,40)Polycyclic presentation:
2^8|b^2=fh,c^2=fgh,d^2=gh,e^2=fg,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 | #133003, #487104, #487232, … |
| maximal normal subgroup of | #133003, #487104, #487232, … |
| maximal quotient of | #585438, #587120, #587558, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51785);