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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1454 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,4)(5,6)(7,8)(9,10,12,14)(11,15,16,13)(19,23)(21,24)(25,26)(27,30)(28,33)(29,32)(31,36)(34,37)(35,39)(38,40);(3,8)(4,7)(9,10,12,14)(11,13,16,15)(17,18,20,22)(19,21,23,24)(26,27)(29,37)(31,33)(35,40);(2,6)(3,8)(17,18)(19,21)(20,22)(23,24)(25,27,30,26)(28,33,36,31)(29,32,37,34)(35,38,40,39);(1,3,5,8)(2,7,6,4)(9,10)(11,13)(12,14)(15,16)(25,28)(26,31)(27,33)(29,35)(30,36)(32,38)(34,39)(37,40);(1,4,5,7)(2,8,6,3)(9,11)(10,13)(12,16)(14,15)(17,19)(18,21)(20,23)(22,24)(25,29,30,37)(26,32,27,34)(28,35,36,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=gi,b^2=fghi,c^2=f,d^2=fg,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, 6122761);