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, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
663 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,16)(12,18)(13,15)(14,17)(19,23)(20,24)(21,22)(25,26,31,34)(27,32,28,29)(30,38,39,33)(35,37,36,40);(3,8)(5,7)(9,11,15,12)(10,16,13,18)(14,20,22,19)(17,24,21,23)(25,27,31,28)(26,29,34,32)(30,36,39,35)(33,40,38,37);(1,3)(2,5)(4,8)(6,7)(9,12,15,11)(10,16,13,18)(14,20,22,19)(17,23,21,24)(25,28,31,27)(26,32,34,29)(30,36,39,35)(33,40,38,37);(1,2)(3,5)(4,6)(7,8)(9,13,15,10)(11,18,12,16)(14,21,22,17)(19,24,20,23)(25,29,31,32)(26,28,34,27)(30,37,39,40)(33,36,38,35);(9,14)(10,17)(11,19)(12,20)(13,21)(15,22)(16,23)(18,24)(25,30,31,39)(26,33,34,38)(27,35,28,36)(29,37,32,40)Polycyclic presentation:
2^8|a^2=fh,b^2=gh,c^2=gh,d^2=gh,e^2=f,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 | #132819, #487538, #488274, … |
| maximal normal subgroup of | #132819, #487538, #488274, … |
| maximal quotient of | #585535, #586018, #587488, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51791);