Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| 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 | |
659 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)(3,6)(4,8)(5,7)(10,14)(11,17)(12,20)(22,24)(25,26,30,31)(27,33,35,28)(29,32,37,38)(34,39,40,36);(1,3,5,8)(2,6,7,4)(26,31)(28,33)(32,38)(36,39);(1,3)(2,4)(5,8)(6,7)(9,10,13,14)(11,18,17,15)(12,16,20,21)(19,24,23,22)(25,27)(26,28)(29,34)(30,35)(31,33)(32,36)(37,40)(38,39);(1,4)(2,3)(5,6)(7,8)(9,11,13,17)(10,15,14,18)(12,19,20,23)(16,22,21,24)(25,28)(26,27)(29,36)(30,33)(31,35)(32,34)(37,39)(38,40);(9,12)(10,16)(11,19)(13,20)(14,21)(15,22)(17,23)(18,24)(25,29,30,37)(26,32,31,38)(27,34,35,40)(28,36,33,39)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=fgh,d^2=fgh,e^2=f,b^a=bf,c^a=cg,d^a=dfg,e^a=efgh,c^b=ch,d^b=df,d^c=dfgh46 conjugacy classes — view the full character table.
| maximal subgroup of | #132802, #487500, #488252, … |
| maximal normal subgroup of | #132802, #487500, #488252, … |
| maximal quotient of | #582192, #582433, #582642, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45942);