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 | |
1696 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,4)(6,7)(9,10)(11,15)(12,16)(13,14)(18,21)(19,23)(25,26,30,28)(27,31,33,34)(29,32,38,36)(35,39,37,40);(2,4)(6,7)(9,11,14,12)(10,15,13,16)(19,23)(22,24)(26,28)(27,33)(32,36)(35,37);(1,2,5,4)(3,6,8,7)(9,10,14,13)(11,16,12,15)(17,18,20,21)(19,22,23,24)(25,27)(26,31)(28,34)(29,37)(30,33)(32,40)(35,38)(36,39);(1,3,5,8)(2,6,4,7)(9,12,14,11)(10,15,13,16)(25,28,30,26)(27,34,33,31)(29,32,38,36)(35,39,37,40);(1,4,5,2)(3,7,8,6)(9,13,14,10)(11,15,12,16)(17,19,20,23)(18,22,21,24)(25,29)(26,32)(27,35)(28,36)(30,38)(31,39)(33,37)(34,40)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=g,d^2=h,e^2=g,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, 3961386);