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 | |
681 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10,14,17)(11,12,15,19)(13,16,21,23)(18,20,22,24)(25,26,30,31)(27,32,35,39)(28,33,34,29)(36,37,38,40);(1,3,5,4)(2,6,8,7)(11,15)(12,19)(13,21)(16,23)(26,31)(27,35)(29,33)(36,38);(1,2)(3,6)(4,7)(5,8)(9,11)(10,12)(13,18)(14,15)(16,20)(17,19)(21,22)(23,24)(25,27)(26,32)(28,38)(29,37)(30,35)(31,39)(33,40)(34,36);(1,4,5,3)(2,7,8,6)(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,28,30,34)(26,33,31,29)(27,36,35,38)(32,37,39,40);(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,29,30,33)(26,34,31,28)(27,37,35,40)(32,36,39,38)Polycyclic presentation:
2^8|a^2=gh,b^2=g,d^2=fg,e^2=f,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×#49014, #7704772, #7704965, … |
| maximal normal subgroup of | C2×#49014, #7704772, #7704965, … |
| maximal quotient of | #561186, #561243, #562409, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 42002);