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 | |
1578 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,13,12)(11,15,16,14)(17,18)(19,24)(20,23)(21,22)(25,26,31,28)(27,29,35,32)(30,36,38,33)(34,39,40,37);(10,12)(11,16)(18,22)(19,23)(25,27,31,35)(26,32,28,29)(30,34,38,40)(33,39,36,37);(10,12)(14,15)(19,23)(20,24)(25,28,31,26)(27,32,35,29)(30,33,38,36)(34,37,40,39);(1,2)(3,5)(4,6)(7,8)(9,11,13,16)(10,14,12,15)(17,19,21,23)(18,20,22,24)(25,29,31,32)(26,27,28,35)(30,37,38,39)(33,34,36,40);(1,3,4,7)(2,5,6,8)(9,12,13,10)(11,15,16,14)(17,20)(18,23)(19,22)(21,24)(25,30)(26,33)(27,34)(28,36)(29,37)(31,38)(32,39)(35,40)Polycyclic presentation:
2^9|a^2=fghi,b^2=h,c^2=h,d^2=f,e^2=fgi,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, 5071620);