Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5024635 |
| 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 | |
1584 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,23)(19,20,24,22)(25,26,30,31)(27,28,34,32)(29,37,40,33)(35,39,38,36);(1,2,5,4)(3,6,7,8)(17,19)(18,20)(21,24)(22,23)(26,31)(27,34)(33,37)(35,38);(9,10,12,14)(11,13,15,16)(17,19,21,24)(18,22,23,20)(25,27,30,34)(26,32,31,28)(29,38,40,35)(33,36,37,39);(1,3)(2,6)(4,8)(5,7)(9,10)(11,13)(12,14)(15,16)(17,19)(18,20)(21,24)(22,23)(25,28,30,32)(26,27,31,34)(29,39,40,36)(33,38,37,35);(1,4,5,2)(3,8,7,6)(9,11,12,15)(10,13,14,16)(17,20)(18,19)(21,22)(23,24)(25,29,30,40)(26,33,31,37)(27,35,34,38)(28,36,32,39)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=g,d^2=f,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^c=dg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4943171);