Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| 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 | |
1709 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,6,5)(3,8,4,7)(9,10,12,15)(11,13,16,14)(19,23)(22,24)(26,30)(27,33)(28,35)(38,40);(1,2,6,5)(3,7,4,8)(9,10)(11,14)(12,15)(13,16)(26,30)(31,36)(32,37)(38,40);(1,3,6,4)(2,7,5,8)(9,11)(10,13)(12,16)(14,15)(17,18,21,20)(19,22,23,24)(25,26,29,30)(27,31,33,36)(28,32,35,37)(34,38,39,40);(1,4,6,3)(2,8,5,7)(9,11)(10,14)(12,16)(13,15)(17,19)(18,22)(20,24)(21,23)(25,27,29,33)(26,31,30,36)(28,34,35,39)(32,38,37,40);(1,5,6,2)(3,8,4,7)(9,10)(11,13)(12,15)(14,16)(17,20,21,18)(19,24,23,22)(25,28)(26,32)(27,34)(29,35)(30,37)(31,38)(33,39)(36,40)Polycyclic presentation:
2^9|a^2=gh,b^2=fgh,c^2=fi,d^2=g,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4359875);