Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1639 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,21)(20,22)(23,24)(25,26,31,28)(27,29,35,32)(30,33,38,36)(34,37,40,39);(10,12)(11,15)(18,19)(20,24)(25,27,31,35)(26,32,28,29)(30,34,38,40)(33,39,36,37);(10,12)(14,16)(17,18,21,19)(20,23,24,22)(25,28,31,26)(27,32,35,29)(30,33,38,36)(34,37,40,39);(1,2)(3,6)(4,5)(7,8)(9,11,13,15)(10,14,12,16)(17,19,21,18)(20,23,24,22)(25,29,31,32)(26,27,28,35)(30,37,38,39)(33,34,36,40);(1,3,4,7)(2,6,5,8)(9,12,13,10)(11,16,15,14)(17,20,21,24)(18,22,19,23)(25,30,31,38)(26,33,28,36)(27,34,35,40)(29,37,32,39)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,c^2=h,d^2=f,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=efh,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, 5270486);