Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6132643 |
| 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 | |
1606 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(3,7)(9,10,13,11)(12,14,16,15)(17,18,22,24)(19,20,21,23)(26,28)(27,33)(29,37)(39,40);(2,5)(6,8)(12,16)(14,15)(19,21)(20,23)(25,26,30,28)(27,31,33,34)(29,36,37,32)(35,40,38,39);(10,11)(14,15)(17,19,22,21)(18,23,24,20)(25,27)(26,31)(28,34)(29,38)(30,33)(32,40)(35,37)(36,39);(1,2,4,5)(3,6,7,8)(9,11,13,10)(12,15,16,14)(17,20,22,23)(18,19,24,21)(25,28,30,26)(27,34,33,31)(29,36,37,32)(35,40,38,39);(1,3)(2,6)(4,7)(5,8)(9,12,13,16)(10,14,11,15)(17,21,22,19)(18,23,24,20)(25,29,30,37)(26,32,28,36)(27,35,33,38)(31,39,34,40)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=h,d^2=fghi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dfi,e^b=eg,d^c=dfg,e^c=ef62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6203938);