Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | |
1592 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)(3,7)(4,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,18,22,20)(19,23,21,24)(25,26,30,27)(28,33,36,31)(29,32,38,34)(35,40,37,39);(1,3,6,4)(2,7,8,5)(9,11,13,16)(10,14,12,15)(17,19,22,21)(18,23,20,24)(26,27)(31,33)(32,34)(39,40);(2,8)(3,4)(10,12)(14,15)(17,20,22,18)(19,23,21,24)(25,27,30,26)(28,31,36,33)(29,34,38,32)(35,39,37,40);(1,4,6,3)(2,7,8,5)(9,12,13,10)(11,15,16,14)(17,19,22,21)(18,24,20,23)(25,28,30,36)(26,31,27,33)(29,37,38,35)(32,40,34,39);(1,5,6,7)(2,3,8,4)(9,11)(10,14)(12,15)(13,16)(17,21,22,19)(18,23,20,24)(25,29,30,38)(26,32,27,34)(28,35,36,37)(31,39,33,40)Polycyclic presentation:
2^9|a^2=gi,b^2=fi,c^2=gi,d^2=i,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5846277);