Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
683 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,7)(3,5)(9,10,13,14)(11,15,19,17)(12,20,21,16)(18,23,24,22)(25,26,30,33)(27,28,35,31)(29,37,38,32)(34,39,40,36);(1,2,6,7)(3,4,5,8)(10,14)(15,17)(16,20)(22,23);(1,3,6,5)(2,8,7,4)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,23)(21,24)(25,27,30,35)(26,31,33,28)(29,34,38,40)(32,39,37,36);(1,4,6,8)(2,3,7,5)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,28,30,31)(26,27,33,35)(29,36,38,39)(32,34,37,40);(1,5,6,3)(2,8,7,4)(9,12,13,21)(10,16,14,20)(11,18,19,24)(15,22,17,23)(25,29)(26,32)(27,34)(28,36)(30,38)(31,39)(33,37)(35,40)Polycyclic presentation:
2^8|a^2=fg,b^2=h,c^2=g,d^2=g,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 | C2×#52819, #8092698, #8092703, … |
| maximal normal subgroup of | C2×#52819, #8092698, #8092703, … |
| maximal quotient of | #582340, #582508, #582552, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45807);