Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4033837 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
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 | |
1600 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(6,8)(9,10,12,14)(11,15,16,13)(19,23)(21,24)(25,26,29,30)(27,31,33,38)(28,35,36,32)(34,40,37,39);(2,5)(6,8)(9,10,12,14)(11,13,16,15)(26,30)(27,33)(28,36)(39,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(17,18,20,22)(19,21,23,24)(25,26)(27,31)(28,35)(29,30)(32,36)(33,38)(34,40)(37,39);(1,3,4,7)(2,6,5,8)(17,19)(18,21)(20,23)(22,24)(25,27,29,33)(26,31,30,38)(28,37,36,34)(32,40,35,39);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(25,28,29,36)(26,32,30,35)(27,34,33,37)(31,39,38,40)Polycyclic presentation:
2^9|a^2=fgh,b^2=gh,c^2=i,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3976038);