Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(11,15)(13,16)(17,18)(19,23)(20,24)(21,22)(25,26,30,31)(27,28,35,32)(29,33,37,38)(34,36,40,39);(2,4)(3,8)(17,19,22,20)(18,23,21,24)(26,31)(28,32)(29,37)(34,40);(1,2,5,4)(3,7,8,6)(9,10,12,14)(11,13,15,16)(17,20,22,19)(18,23,21,24)(25,27,30,35)(26,32,31,28)(29,34,37,40)(33,39,38,36);(1,3,5,8)(2,6,4,7)(9,11,12,15)(10,13,14,16)(17,21,22,18)(19,23,20,24)(25,28,30,32)(26,27,31,35)(29,36,37,39)(33,34,38,40);(1,4,5,2)(3,7,8,6)(25,29)(26,33)(27,34)(28,36)(30,37)(31,38)(32,39)(35,40)Polycyclic presentation:
2^9|a^2=fh,b^2=fg,c^2=i,d^2=i,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4431535);