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, cosocle | C23 | C25 |
| prime radical | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
673 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)(3,7)(4,6)(5,8)(9,10)(11,15)(12,14)(13,20)(16,21)(17,23)(18,19)(22,24)(25,26,30,33)(27,28,35,31)(29,37,38,32)(34,39,40,36);(1,2,4,6)(3,7,8,5)(10,12)(15,19)(16,20)(22,23);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,23)(21,24)(25,27,30,35)(26,31,33,28)(29,34,38,40)(32,39,37,36);(1,3,4,8)(2,5,6,7)(9,12,14,10)(11,15,18,19)(13,20,21,16)(17,22,24,23)(25,28,30,31)(26,27,33,35)(29,36,38,39)(32,34,37,40);(9,13,14,21)(10,16,12,20)(11,17,18,24)(15,22,19,23)(25,29,30,38)(26,32,33,37)(27,34,35,40)(28,36,31,39)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^2=g,d^2=fg,e^2=fgh,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 | #7221596, #7221600, #7221660, … |
| maximal normal subgroup of | #7221596, #7221600, #7221660, … |
| maximal quotient of | #582291, #582493, #582533, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 46080);