Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| 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 | |
672 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(5,8)(9,10,14,15)(11,17,18,12)(13,21,22,16)(19,20,24,23)(25,26,30,33)(27,31,34,29)(28,32,36,40)(35,39,37,38);(1,2,4,6)(3,5,7,8)(10,15)(11,18)(13,22)(20,23)(27,34)(28,36)(29,31)(32,40);(1,3,4,7)(2,5,6,8)(9,11,14,18)(10,12,15,17)(13,19,22,24)(16,20,21,23)(25,27)(26,31)(28,37)(29,33)(30,34)(32,38)(35,36)(39,40);(9,12)(10,11)(13,20)(14,17)(15,18)(16,19)(21,24)(22,23)(25,28,30,36)(26,32,33,40)(27,35,34,37)(29,38,31,39);(9,13,14,22)(10,16,15,21)(11,19,18,24)(12,20,17,23)(25,29,30,31)(26,27,33,34)(28,38,36,39)(32,35,40,37)Polycyclic presentation:
2^8|a^2=h,b^2=fg,c^2=g,d^2=fh,e^2=h,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#52068, #7708132, #7708136, … |
| maximal normal subgroup of | C2×#52068, #7708132, #7708136, … |
| maximal quotient of | #560366, #560538, #561548, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45056);