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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1778 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)(3,4)(5,7)(6,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,26,30,31)(27,35,37,32)(28,33,38,36)(29,39,40,34);(1,3)(2,6)(4,7)(5,8)(9,11,13,16)(10,12,14,15)(18,21)(22,23)(26,31)(28,38)(32,35)(34,39);(2,7)(3,8)(10,14)(11,16)(17,18)(19,23)(20,21)(22,24)(25,27,30,37)(26,32,31,35)(28,34,38,39)(29,33,40,36);(1,4)(2,3)(5,6)(7,8)(9,12)(10,11)(13,15)(14,16)(25,28)(26,33)(27,34)(29,35)(30,38)(31,36)(32,40)(37,39);(1,3)(2,4)(5,8)(6,7)(9,10)(11,12)(13,14)(15,16)(17,19)(18,22)(20,24)(21,23)(25,29,30,40)(26,34,31,39)(27,36,37,33)(28,35,38,32)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=fh,e^2=fh,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, 4533361);