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) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2116 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)(3,4,8,7)(9,10,14,16)(11,13,12,15)(18,20)(19,23)(25,26,29,32)(27,33,35,30)(28,31,36,38)(34,39,40,37);(2,6)(3,8)(9,11)(10,15)(12,14)(13,16)(17,18,21,20)(19,24,23,22)(28,36)(31,38)(34,40)(37,39);(3,8)(4,7)(11,12)(13,15)(18,20)(19,23)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,38)(37,40);(1,3,5,8)(2,7,6,4)(9,12,14,11)(10,15,16,13)(17,19,21,23)(18,22,20,24)(25,27,29,35)(26,30,32,33)(28,34,36,40)(31,37,38,39);(1,4,5,7)(2,8,6,3)(9,13,14,15)(10,11,16,12)(17,20,21,18)(19,24,23,22)(25,28)(26,31)(27,34)(29,36)(30,37)(32,38)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=hi,b^2=fghi,d^2=fg,e^2=f,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, 4972404);