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 | |
671 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,7)(3,8,4,6)(25,26)(27,29)(28,35)(30,32)(31,33)(34,37)(36,39)(38,40);(1,3)(2,6)(4,5)(7,8)(10,14)(12,20)(15,17)(18,24)(25,27,31,28)(26,29,33,35)(30,36,38,34)(32,39,40,37);(3,4)(6,8)(9,10,13,14)(11,17,19,15)(12,16,20,21)(18,23,24,22)(25,28,31,27)(26,29,33,35)(30,36,38,34)(32,37,40,39);(1,4,5,3)(2,6,7,8)(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,29)(26,28)(27,33)(30,37)(31,35)(32,36)(34,40)(38,39);(1,2)(3,8)(4,6)(5,7)(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,30,31,38)(26,32,33,40)(27,34,28,36)(29,37,35,39)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=fgh,d^2=h,e^2=g,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×#58663, #8104785, #8104842, … |
| maximal normal subgroup of | C2×#58663, #8104785, #8104842, … |
| maximal quotient of | #586240, #586398, #587166, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51651);