Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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 | |
1446 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)(3,7)(4,6)(5,8)(11,15)(14,16)(17,18,20,22)(19,23,24,21)(25,26,30,27)(28,33,36,31)(29,34,38,32)(35,39,37,40);(1,3,6,5)(2,7,4,8)(10,13)(14,16)(26,27)(31,33)(32,34)(39,40);(1,2,6,4)(3,8,5,7)(9,10)(11,14)(12,13)(15,16)(25,27,30,26)(28,33,36,31)(29,34,38,32)(35,40,37,39);(1,4,6,2)(3,7,5,8)(17,19,20,24)(18,21,22,23)(25,28,30,36)(26,31,27,33)(29,37,38,35)(32,40,34,39);(1,5,6,3)(2,7,4,8)(9,11,12,15)(10,14,13,16)(25,29,30,38)(26,32,27,34)(28,35,36,37)(31,39,33,40)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=fg,d^2=i,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4802833);