Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4033837 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1908 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)(3,7,8,5)(11,16)(13,15)(17,18,20,22)(19,23,24,21)(25,26,29,30)(27,33,35,31)(28,36,37,32)(34,39,38,40);(3,8)(5,7)(9,10,14,12)(11,13,16,15)(17,18,20,22)(19,21,24,23)(26,30)(28,37)(31,33)(34,38);(1,2)(3,5)(4,6)(7,8)(9,11,14,16)(10,15,12,13)(17,19)(18,21)(20,24)(22,23)(25,27,29,35)(26,31,30,33)(28,38,37,34)(32,40,36,39);(1,2)(3,5)(4,6)(7,8)(9,12,14,10)(11,13,16,15)(25,26)(27,31)(28,36)(29,30)(32,37)(33,35)(34,40)(38,39);(1,3)(2,5)(4,8)(6,7)(9,13)(10,11)(12,16)(14,15)(17,18)(19,21)(20,22)(23,24)(25,28)(26,32)(27,34)(29,37)(30,36)(31,39)(33,40)(35,38)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=fh,d^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3964445);