Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6132643 |
| 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 | |
1477 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,15)(12,16)(13,14)(19,23)(22,24)(25,26,29,33)(27,30,35,31)(28,36,38,32)(34,40,37,39);(3,8)(5,7)(10,14)(11,16)(18,21)(19,23)(25,27,29,35)(26,30,33,31)(28,37,38,34)(32,40,36,39);(11,16)(12,15)(17,18,20,21)(19,22,23,24)(25,26)(27,30)(28,32)(29,33)(31,35)(34,39)(36,38)(37,40);(1,3)(2,5)(4,8)(6,7)(9,11,13,16)(10,15,14,12)(17,19,20,23)(18,22,21,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,40)(36,39)(37,38);(1,2)(3,5)(4,6)(7,8)(9,12,13,15)(10,16,14,11)(25,28,29,38)(26,32,33,36)(27,34,35,37)(30,39,31,40)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=h,d^2=fh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dfi,e^b=eg,d^c=dfg,e^c=ef62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6082196);