Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
674 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10,14,13)(11,15,19,18)(12,20,22,16)(17,24,21,23)(25,26,30,33)(27,31,34,29)(28,32,38,40)(35,39,36,37);(1,3,5,4)(2,6,8,7)(10,13)(15,18)(16,20)(23,24)(27,34)(29,31)(35,36)(37,39);(1,2)(3,6)(4,7)(5,8)(9,11,14,19)(10,15,13,18)(12,21,22,17)(16,24,20,23)(25,27)(26,31)(28,36)(29,33)(30,34)(32,37)(35,38)(39,40);(1,4,5,3)(2,7,8,6)(9,12,14,22)(10,16,13,20)(11,17,19,21)(15,23,18,24)(25,28)(26,32)(27,35)(29,37)(30,38)(31,39)(33,40)(34,36);(9,13,14,10)(11,18,19,15)(12,20,22,16)(17,24,21,23)(25,29,30,31)(26,27,33,34)(28,37,38,39)(32,35,40,36)Polycyclic presentation:
2^8|a^2=fgh,b^2=g,c^2=f,d^2=fg,e^2=fh,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #6702494, #6702499, #6702559, … |
| maximal normal subgroup of | #6702494, #6702499, #6702559, … |
| maximal quotient of | #544658, #545181, #545620, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 35736);