Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
1565 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)(10,11)(12,16)(17,18,21,19)(20,23,24,22)(25,26,29,30)(27,31,34,38)(28,32,37,36)(33,39,35,40);(1,2,5,4)(3,6,7,8)(9,10,13,11)(12,14,16,15)(26,30)(31,38)(32,36)(39,40);(2,4)(6,8)(9,10,13,11)(12,15,16,14)(17,19,21,18)(20,23,24,22)(25,27,29,34)(26,31,30,38)(28,35,37,33)(32,40,36,39);(1,3)(2,6)(4,8)(5,7)(9,11,13,10)(12,14,16,15)(17,20,21,24)(18,22,19,23)(25,26)(27,31)(28,36)(29,30)(32,37)(33,40)(34,38)(35,39);(1,4,5,2)(3,6,7,8)(9,12,13,16)(10,14,11,15)(25,28,29,37)(26,32,30,36)(27,33,34,35)(31,39,38,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=i,d^2=fgi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3454261);