Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1664 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,7)(3,6,8,5)(10,13)(11,15)(17,18,20,21)(19,22,23,24)(25,26,29,32)(27,33,34,30)(28,31,36,39)(35,40,37,38);(1,3)(2,5)(4,8)(6,7)(9,10,12,13)(11,16,15,14)(18,21)(19,23)(27,34)(28,36)(30,33)(31,39);(3,8)(5,6)(11,15)(14,16)(17,18,20,21)(19,22,23,24)(25,27,29,34)(26,30,32,33)(28,37,36,35)(31,40,39,38);(1,3)(2,6)(4,8)(5,7)(9,11)(10,14)(12,15)(13,16)(17,18)(19,22)(20,21)(23,24)(25,28,29,36)(26,31,32,39)(27,35,34,37)(30,38,33,40);(1,2)(3,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,22)(20,23)(21,24)(25,26)(27,30)(28,31)(29,32)(33,34)(35,38)(36,39)(37,40)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,c^2=fhi,d^2=g,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, 5926660);