Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53158 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| 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 | |
665 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(3,4)(11,17)(15,20)(18,23)(22,24)(25,26,29,32)(27,30,34,37)(28,31,35,38)(33,36,39,40);(1,2,6,5)(3,8,4,7)(10,14)(12,19)(15,20)(18,23)(28,35)(31,38)(33,39)(36,40);(1,3,6,4)(2,7,5,8)(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24)(25,26)(27,30)(28,31)(29,32)(33,36)(34,37)(35,38)(39,40);(1,4,6,3)(2,8,5,7)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,27,29,34)(26,30,32,37)(28,33,35,39)(31,36,38,40);(1,5,6,2)(3,8,4,7)(9,12)(10,16)(11,18)(13,19)(14,21)(15,22)(17,23)(20,24)(25,28)(26,31)(27,33)(29,35)(30,36)(32,38)(34,39)(37,40)Polycyclic presentation:
2^8|a^2=fg,b^2=f,c^2=h,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=df,e^b=egh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#53327, #8105852, #8105897, … |
| maximal normal subgroup of | C2×#53327, #8105852, #8105897, … |
| maximal quotient of | #606808, #606839, #606848, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 46315);