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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1384 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,7,5)(3,8,6,4)(11,15)(14,16)(17,18,20,22)(19,23,24,21)(25,26)(27,33)(28,36)(29,30)(31,35)(32,38)(34,39)(37,40);(1,3,7,6)(2,8,5,4)(10,13)(14,16)(26,30)(31,33)(32,36)(39,40);(1,4)(2,6)(3,5)(7,8)(9,10,12,13)(11,14,15,16)(25,26)(27,31)(28,32)(29,30)(33,35)(34,39)(36,38)(37,40);(1,5,7,2)(3,8,6,4)(17,19,20,24)(18,21,22,23)(25,27,29,35)(26,31,30,33)(28,37,38,34)(32,40,36,39);(1,6,7,3)(2,8,5,4)(9,11)(10,14)(12,15)(13,16)(25,28,29,38)(26,32,30,36)(27,34,35,37)(31,39,33,40)Polycyclic presentation:
2^9|a^2=fgi,b^2=g,c^2=fgh,d^2=gi,e^2=fg,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, 4802663);