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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1656 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(17,18,20,21)(19,22,24,23)(25,26,30,31)(27,29,35,32)(28,36,38,33)(34,39,40,37);(3,7)(6,8)(11,16)(13,15)(17,18,20,21)(19,23,24,22)(26,31)(29,32)(33,36)(37,39);(2,5)(6,8)(18,21)(22,23)(25,27,30,35)(26,32,31,29)(28,34,38,40)(33,39,36,37);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(17,18)(19,22)(20,21)(23,24)(25,28,30,38)(26,33,31,36)(27,34,35,40)(29,37,32,39);(1,3)(2,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,22)(20,24)(21,23)(25,29,30,32)(26,27,31,35)(28,37,38,39)(33,34,36,40)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=f,d^2=f,e^2=f,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, 6071578);