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, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
662 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,6)(7,8)(9,10,14,18)(11,13,17,15)(12,16,20,24)(19,22,21,23)(25,26)(27,34)(28,36)(29,30)(31,33)(32,39)(35,38)(37,40);(11,17)(12,20)(13,15)(16,24)(25,27,31,29)(26,30,33,34)(28,37,39,35)(32,40,36,38);(1,2)(3,7)(4,8)(5,6)(9,11,14,17)(10,15,18,13)(12,21,20,19)(16,22,24,23)(25,28,31,39)(26,32,33,36)(27,35,29,37)(30,38,34,40);(1,3,5,4)(2,7,6,8)(9,12,14,20)(10,16,18,24)(11,19,17,21)(13,22,15,23)(25,29,31,27)(26,30,33,34)(28,35,39,37)(32,40,36,38);(1,4,5,3)(2,8,6,7)(9,13,14,15)(10,17,18,11)(12,22,20,23)(16,21,24,19)(25,30)(26,27)(28,38)(29,33)(31,34)(32,35)(36,37)(39,40)Polycyclic presentation:
2^8|a^2=fh,b^2=f,c^2=h,d^2=g,e^2=fg,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×#50517, #7706031, #7706230, … |
| maximal normal subgroup of | C2×#50517, #7706031, #7706230, … |
| maximal quotient of | #560772, #561042, #561892, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 43505);