Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| 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, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
660 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10,13,14)(11,17,19,15)(12,20,21,16)(18,22,24,23)(28,36)(31,38)(34,40)(37,39);(10,14)(15,17)(16,20)(22,23)(25,26,29,32)(27,33,35,30)(28,31,36,38)(34,39,40,37);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,23)(21,24)(25,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,38)(39,40);(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,27,29,35)(26,30,32,33)(28,34,36,40)(31,37,38,39);(9,12,13,21)(10,16,14,20)(11,18,19,24)(15,22,17,23)(25,28)(26,31)(27,34)(29,36)(30,37)(32,38)(33,39)(35,40)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^2=fg,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#37197, #7634605, #7634608, … |
| maximal normal subgroup of | C2×#37197, #7634605, #7634608, … |
| maximal quotient of | #553802, #553807, #554046, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30185);