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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1510 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)(10,15)(11,12)(19,23)(22,24)(25,26)(27,31)(28,36)(29,30)(32,37)(33,34)(35,40)(38,39);(3,8)(5,7)(9,10,14,15)(11,13,12,16)(18,21)(22,24)(26,30)(27,33)(32,36)(35,39);(9,11,14,12)(10,16,15,13)(17,18,20,21)(19,22,23,24)(25,26)(27,34)(28,36)(29,30)(31,33)(32,37)(35,38)(39,40);(1,3)(2,5)(4,8)(6,7)(9,12,14,11)(10,16,15,13)(25,27,29,33)(26,31,30,34)(28,35,37,39)(32,38,36,40);(1,2)(3,5)(4,6)(7,8)(9,13,14,16)(10,11,15,12)(17,19,20,23)(18,22,21,24)(25,28,29,37)(26,32,30,36)(27,35,33,39)(31,38,34,40)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=fgh,d^2=f,e^2=h,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, 6087599);