Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
1640 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)(11,15)(12,14)(13,16)(18,21)(19,23)(25,26,30,31)(27,32,35,39)(28,36,37,29)(33,34,40,38);(9,11,14,13)(10,15,12,16)(19,23)(22,24)(26,31)(28,37)(32,39)(33,40);(9,10,14,12)(11,16,13,15)(17,18)(19,22)(20,21)(23,24)(25,27)(26,32)(28,33)(29,38)(30,35)(31,39)(34,36)(37,40);(1,2)(3,5)(4,6)(7,8)(9,12,14,10)(11,15,13,16)(17,19)(18,22)(20,23)(21,24)(25,28,30,37)(26,29,31,36)(27,33,35,40)(32,38,39,34);(1,3,4,7)(2,5,6,8)(9,13,14,11)(10,15,12,16)(17,18)(19,22)(20,21)(23,24)(25,29)(26,28)(27,34)(30,36)(31,37)(32,40)(33,39)(35,38)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=h,d^2=fh,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3567137);