Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 | |
2123 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,3,4)(9,10)(11,14)(12,15)(13,18)(16,21)(17,19)(20,23)(22,24)(25,26,31,28)(27,34,32,29)(30,33,39,36)(35,40,38,37);(6,8)(10,11)(12,19)(13,21)(22,23)(25,27,31,32)(26,29,28,34)(30,38,39,35)(33,40,36,37);(5,6,7,8)(9,11,14,10)(12,15,19,17)(13,18,21,16)(20,22,24,23)(25,28,31,26)(27,29,32,34)(30,36,39,33)(35,37,38,40);(1,2)(3,4)(9,12,14,19)(10,15,11,17)(13,20,21,24)(16,22,18,23)(25,29)(26,32)(27,28)(30,37)(31,34)(33,38)(35,36)(39,40);(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,30,31,39)(26,33,28,36)(27,35,32,38)(29,37,34,40)Polycyclic presentation:
2^9|a^2=i,b^2=fg,c^2=h,d^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1690396);