Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1898 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,6)(3,4,7,8)(11,15)(14,16)(19,23)(22,24)(25,26)(27,31)(28,32)(29,30)(33,38)(34,37)(35,36)(39,40);(1,3,5,7)(2,4,6,8)(10,13)(14,16)(18,21)(22,24)(26,30)(31,37)(32,35)(38,39);(1,4)(2,3)(5,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,18,20,21)(19,22,23,24)(25,26)(27,31)(28,35)(29,30)(32,36)(33,39)(34,37)(38,40);(1,3)(2,4)(5,7)(6,8)(17,19)(18,22)(20,23)(21,24)(25,27,29,34)(26,31,30,37)(28,33,36,40)(32,38,35,39);(1,2)(3,4)(5,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,19)(18,22)(20,23)(21,24)(25,28)(26,32)(27,33)(29,36)(30,35)(31,38)(34,40)(37,39)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=i,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4538564);