Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| 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 | |
1456 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,5)(3,7,8,6)(9,10)(11,15)(12,16)(13,14)(18,21)(19,23)(27,33)(30,34)(35,39)(37,40);(2,5)(3,8)(10,14)(11,16)(18,21)(22,24)(25,26,29,32)(27,34,33,30)(28,31,36,38)(35,40,39,37);(3,8)(6,7)(9,11)(10,12)(13,16)(14,15)(17,18,20,21)(19,24,23,22)(25,26)(27,30)(28,31)(29,32)(33,34)(35,37)(36,38)(39,40);(1,3)(2,6)(4,8)(5,7)(9,12)(10,11)(13,15)(14,16)(17,19,20,23)(18,22,21,24)(25,27)(26,30)(28,35)(29,33)(31,37)(32,34)(36,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,12)(13,14)(15,16)(25,28,29,36)(26,31,32,38)(27,35,33,39)(30,37,34,40)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=fghi,d^2=fghi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6038811);