Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| 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 | |
1552 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)(3,7)(4,5)(6,8)(11,15)(14,16)(25,26,29,32)(27,33,34,30)(28,31,36,38)(35,39,40,37);(1,3,5,8)(2,6,4,7)(9,10,13,12)(11,14,15,16)(18,21)(22,24)(27,34)(28,36)(30,33)(31,38);(3,8)(6,7)(17,18)(19,22)(20,21)(23,24)(25,27,29,34)(26,30,32,33)(28,35,36,40)(31,37,38,39);(1,3)(2,7)(4,6)(5,8)(9,11)(10,14)(12,16)(13,15)(25,28)(26,31)(27,35)(29,36)(30,37)(32,38)(33,39)(34,40);(1,4,5,2)(3,7,8,6)(9,12,13,10)(11,16,15,14)(17,19,20,23)(18,22,21,24)(25,26)(27,30)(28,31)(29,32)(33,34)(35,37)(36,38)(39,40)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=g,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2705227);