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)(3,5)(4,7)(6,8)(9,10,14,17)(11,13,16,22)(12,15,21,24)(18,20,19,23)(25,26,30,33)(27,34,31,28)(29,32,38,37)(35,40,39,36);(2,3)(6,7)(11,16)(13,22)(18,19)(20,23)(25,26,30,33)(27,28,31,34)(29,37,38,32)(35,40,39,36);(1,2,5,3)(4,6,8,7)(9,11,14,16)(10,13,17,22)(12,19,21,18)(15,23,24,20)(25,27)(26,28)(29,35)(30,31)(32,36)(33,34)(37,40)(38,39);(1,3,5,2)(4,7,8,6)(9,12)(10,15)(11,18)(13,20)(14,21)(16,19)(17,24)(22,23)(25,28)(26,31)(27,33)(29,36)(30,34)(32,39)(35,37)(38,40);(1,4,5,8)(2,6,3,7)(9,13)(10,16)(11,17)(12,20)(14,22)(15,19)(18,24)(21,23)(25,29,30,38)(26,32,33,37)(27,35,31,39)(28,36,34,40)Polycyclic presentation:
2^8|a^2=gh,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×#56892, #8099980, #8100313, … |
| maximal normal subgroup of | C2×#56892, #8099980, #8100313, … |
| maximal quotient of | #585774, #586552, #586999, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 49880);