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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
709 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)(3,7)(4,5)(6,8)(9,10,14,17)(11,12,19,15)(13,16,21,23)(18,20,24,22)(26,30)(27,33)(28,36)(38,40);(2,5)(6,7)(13,21)(16,23)(18,24)(20,22)(25,26,29,30)(27,34,33,31)(28,32,36,37)(35,40,39,38);(9,11,14,19)(10,15,17,12)(13,18,21,24)(16,22,23,20)(25,26)(27,31)(28,32)(29,30)(33,34)(35,38)(36,37)(39,40);(1,2)(3,6)(4,5)(7,8)(9,12,14,15)(10,11,17,19)(13,20,21,22)(16,18,23,24)(25,27,29,33)(26,31,30,34)(28,35,36,39)(32,38,37,40);(1,3)(2,6)(4,8)(5,7)(9,13,14,21)(10,16,17,23)(11,18,19,24)(12,20,15,22)(25,28,29,36)(26,32,30,37)(27,35,33,39)(31,38,34,40)Polycyclic presentation:
2^8|a^2=gh,b^2=h,c^2=gh,d^2=g,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×#57921, #8101648, #8101956, … |
| maximal normal subgroup of | C2×#57921, #8101648, #8101956, … |
| maximal quotient of | #586384, #586659, #586865, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 50909);