Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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 | |
1620 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,6)(4,8)(9,10,12,14)(11,15,16,13)(17,18,21,19)(20,23,24,22)(25,26)(27,33)(28,36)(29,30)(31,34)(32,37)(35,38)(39,40);(2,6)(3,7)(11,16)(13,15)(17,18,21,19)(20,22,24,23)(26,30)(27,34)(32,36)(35,40);(1,2,5,6)(3,4,7,8)(9,10)(11,13)(12,14)(15,16)(17,19,21,18)(20,23,24,22)(25,27)(26,31)(28,35)(29,34)(30,33)(32,38)(36,39)(37,40);(1,3,5,7)(2,4,6,8)(9,11)(10,13)(12,16)(14,15)(17,20)(18,22)(19,23)(21,24)(25,26)(27,31)(28,36)(29,30)(32,37)(33,34)(35,39)(38,40);(1,4)(2,7)(3,6)(5,8)(25,28,29,37)(26,32,30,36)(27,35,34,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,c^2=fgi,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4796032);