Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| 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 | |
1529 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,6,5)(3,7,4,8)(11,15)(14,16)(17,18,20,22)(19,21,23,24)(25,26)(27,30)(28,33)(29,32)(31,36)(34,37)(35,39)(38,40);(1,2)(3,8)(4,7)(5,6)(10,13)(14,16)(17,19,20,23)(18,21,22,24)(26,27)(31,33)(32,34)(38,39);(1,3,6,4)(2,7,5,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,21)(20,23)(22,24)(25,27,30,26)(28,31,36,33)(29,32,37,34)(35,39,40,38);(1,4,6,3)(2,7,5,8)(17,18)(19,21)(20,22)(23,24)(25,28,30,36)(26,31,27,33)(29,35,37,40)(32,38,34,39);(1,5,6,2)(3,7,4,8)(9,11)(10,14)(12,15)(13,16)(17,18)(19,21)(20,22)(23,24)(25,29)(26,32)(27,34)(28,35)(30,37)(31,38)(33,39)(36,40)Polycyclic presentation:
2^9|a^2=gi,b^2=fi,c^2=f,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5393631);