Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| 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 | |
667 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)(11,17)(12,14)(13,20)(15,19)(16,21)(18,22)(23,24)(25,26,30,29)(27,33,34,31)(28,36,37,32)(35,38,40,39);(1,2,4,6)(3,5,8,7)(10,12)(13,21)(15,17)(18,24)(27,34)(31,33)(35,40)(38,39);(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,27,30,34)(26,31,29,33)(28,35,37,40)(32,38,36,39);(1,2)(3,5)(4,6)(7,8)(9,12,14,10)(11,15,19,17)(13,20,21,16)(18,22,24,23)(25,28,30,37)(26,32,29,36)(27,35,34,40)(31,38,33,39);(1,3,4,8)(2,5,6,7)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,29,30,26)(27,33,34,31)(28,36,37,32)(35,39,40,38)Polycyclic presentation:
2^8|a^2=h,b^2=fgh,c^2=g,d^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df,e^b=egh,d^c=dgh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#56254, #8099917, #8100300, … |
| maximal normal subgroup of | C2×#56254, #8099917, #8100300, … |
| maximal quotient of | #585769, #586205, #586609, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 49242);