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 | |
1506 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,6,8)(3,5,4,7)(9,10)(11,14)(12,15)(13,16)(18,21)(22,24)(25,26,29,32)(27,30,35,33)(28,31,36,38)(34,37,40,39);(1,3)(2,7)(4,6)(5,8)(9,10,13,16)(11,12,15,14)(18,21)(19,23)(28,36)(31,38)(34,40)(37,39);(3,4)(5,7)(11,15)(12,14)(17,18,20,21)(19,22,23,24)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,38)(37,40);(1,4,6,3)(2,7,8,5)(9,11,13,15)(10,14,16,12)(17,18)(19,22)(20,21)(23,24)(25,27)(26,30)(28,34)(29,35)(31,37)(32,33)(36,40)(38,39);(1,5,6,7)(2,3,8,4)(9,12,13,14)(10,15,16,11)(17,19,20,23)(18,22,21,24)(25,28,29,36)(26,31,32,38)(27,34,35,40)(30,37,33,39)Polycyclic presentation:
2^9|a^2=gh,b^2=fhi,c^2=i,d^2=f,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, 6176706);