Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| 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 | |
672 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(6,8)(12,19)(16,21)(18,23)(22,24)(25,26,30,33)(27,28,35,31)(29,32,37,39)(34,36,40,38);(10,14)(11,17)(16,21)(18,23)(25,27,30,35)(26,31,33,28)(29,34,37,40)(32,38,39,36);(1,2,4,5)(3,6,7,8)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,27)(26,28)(29,34)(30,35)(31,33)(32,36)(37,40)(38,39);(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24)(25,28)(26,27)(29,36)(30,31)(32,34)(33,35)(37,38)(39,40);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,18)(13,19)(14,21)(15,22)(17,23)(20,24)(25,29,30,37)(26,32,33,39)(27,34,35,40)(28,36,31,38)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=g,d^2=fh,e^2=fg,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#37180, #7634480, #7634503, … |
| maximal normal subgroup of | C2×#37180, #7634480, #7634503, … |
| maximal quotient of | #553881, #553887, #554099, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30168);