Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1566 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,6,5)(3,8,7,4)(11,15)(12,14)(19,23)(21,24)(25,26)(27,33)(28,36)(29,34)(30,31)(32,37)(35,39)(38,40);(1,3,6,7)(2,4,5,8)(9,10,13,16)(11,12,15,14)(17,18,20,22)(19,21,23,24)(26,31)(27,34)(28,37)(38,39);(1,4)(2,7)(3,5)(6,8)(9,11,13,15)(10,14,16,12)(25,27)(26,29)(28,35)(30,34)(31,33)(32,39)(36,38)(37,40);(1,4)(2,3)(5,7)(6,8)(9,10)(11,14)(12,15)(13,16)(17,19)(18,21)(20,23)(22,24)(25,28)(26,32)(27,35)(29,39)(30,37)(31,36)(33,38)(34,40);(1,5,6,2)(3,8,7,4)(9,12,13,14)(10,15,16,11)(25,29)(26,27)(28,38)(30,33)(31,34)(32,40)(35,36)(37,39)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=fg,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4781086);