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 | |
1745 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,8)(3,4,7,6)(11,15)(14,16)(17,18,20,22)(19,23,24,21)(25,26)(27,31)(28,32)(29,30)(33,38)(34,37)(35,36)(39,40);(3,7)(4,6)(10,13)(14,16)(17,18,20,22)(19,21,24,23)(26,30)(31,37)(32,35)(38,39);(1,3,5,7)(2,6,8,4)(9,10,12,13)(11,14,15,16)(17,19)(18,21)(20,24)(22,23)(25,26)(27,31)(28,35)(29,30)(32,36)(33,39)(34,37)(38,40);(1,4,5,6)(2,7,8,3)(17,18)(19,21)(20,22)(23,24)(25,27,29,34)(26,31,30,37)(28,33,36,40)(32,38,35,39);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,15)(13,16)(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=i,c^2=fh,d^2=fgi,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, 4590600);