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 | |
680 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(9,10,14,15)(11,13,19,16)(12,20,21,17)(18,24,22,23)(27,31)(28,35)(34,39)(36,40);(1,2)(3,7)(4,6)(5,8)(10,15)(12,21)(13,16)(18,22)(25,26,30,33)(27,28,31,35)(29,37,38,32)(34,40,39,36);(1,2,4,6)(3,5,8,7)(9,11)(10,16)(12,22)(13,15)(14,19)(17,23)(18,21)(20,24)(25,27)(26,28)(29,34)(30,31)(32,36)(33,35)(37,40)(38,39);(1,2)(3,5)(4,6)(7,8)(9,12,14,21)(10,17,15,20)(11,18,19,22)(13,23,16,24)(25,28)(26,31)(27,33)(29,36)(30,35)(32,39)(34,37)(38,40);(1,3)(2,5)(4,8)(6,7)(9,13,14,16)(10,11,15,19)(12,23,21,24)(17,18,20,22)(25,29,30,38)(26,32,33,37)(27,34,31,39)(28,36,35,40)Polycyclic presentation:
2^8|a^2=f,b^2=fg,c^2=h,d^2=f,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×#56520, #8099119, #8099263, … |
| maximal normal subgroup of | C2×#56520, #8099119, #8099263, … |
| maximal quotient of | #585706, #586487, #586999, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 49508);