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 | |
1452 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)(3,4)(5,6)(7,8)(9,10)(11,16)(12,14)(13,15)(19,23)(21,24)(25,26,31,35)(27,32,28,38)(29,30,34,33)(36,40,39,37);(2,6)(3,7)(9,10,13,15)(11,12,14,16)(25,27,31,28)(26,32,35,38)(29,39,34,36)(30,37,33,40);(1,2,5,6)(3,4,7,8)(9,11,13,14)(10,12,15,16)(17,18,20,22)(19,21,23,24)(25,28,31,27)(26,32,35,38)(29,39,34,36)(30,40,33,37);(1,3)(2,4)(5,7)(6,8)(9,10)(11,12)(13,15)(14,16)(17,19)(18,21)(20,23)(22,24)(25,29,31,34)(26,33,35,30)(27,36,28,39)(32,37,38,40);(1,4)(2,7)(3,6)(5,8)(9,12)(10,14)(11,15)(13,16)(25,30,31,33)(26,34,35,29)(27,37,28,40)(32,36,38,39)Polycyclic presentation:
2^9|a^2=fg,b^2=gh,c^2=fi,d^2=fg,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, 4786465);