Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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 | |
1621 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)(10,14)(11,15)(17,18,20,21)(19,23,24,22)(25,26,30,33)(27,28,35,31)(29,32,37,39)(34,36,40,38);(2,6)(3,7)(9,10,13,14)(11,16,15,12)(18,21)(22,23)(29,37)(32,39)(34,40)(36,38);(3,7)(4,8)(11,15)(12,16)(17,18,20,21)(19,23,24,22)(25,27,30,35)(26,31,33,28)(29,34,37,40)(32,38,39,36);(1,3,5,7)(2,4,6,8)(9,11)(10,12)(13,15)(14,16)(17,18)(19,22)(20,21)(23,24)(25,28,30,31)(26,27,33,35)(29,36,37,38)(32,34,39,40);(1,4)(2,7)(3,6)(5,8)(9,12)(10,11)(13,16)(14,15)(17,19,20,24)(18,22,21,23)(25,29,30,37)(26,32,33,39)(27,34,35,40)(28,36,31,38)Polycyclic presentation:
2^9|a^2=fhi,b^2=ghi,c^2=gh,d^2=fi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5026788);