Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | |
1497 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(3,7)(9,10)(11,12)(13,15)(14,16)(19,23)(21,24)(25,26,30,27)(28,33,36,31)(29,32,37,34)(35,39,40,38);(1,2,4,5)(3,6,7,8)(9,10,13,15)(11,12,16,14)(26,27)(31,33)(32,34)(38,39);(1,3,4,7)(2,6,5,8)(9,11,13,16)(10,14,15,12)(17,18,20,22)(19,21,23,24)(25,27,30,26)(28,33,36,31)(29,32,37,34)(35,38,40,39);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,14)(15,16)(17,19,20,23)(18,21,22,24)(25,28)(26,31)(27,33)(29,35)(30,36)(32,38)(34,39)(37,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,12)(13,16)(14,15)(25,29,30,37)(26,32,27,34)(28,35,36,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=fghi,d^2=ghi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4611793);