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) = C25×C4 |
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 | |
1519 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,12,13)(11,14,15,16)(18,22)(19,23)(28,35)(32,37)(33,39)(38,40);(11,15)(14,16)(17,18,21,22)(19,24,23,20)(26,30)(31,36)(32,37)(38,40);(10,13)(14,16)(17,19,21,23)(18,20,22,24)(25,26,29,30)(27,31,34,36)(28,32,35,37)(33,38,39,40);(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,20)(18,19)(21,24)(22,23)(25,27,29,34)(26,31,30,36)(28,33,35,39)(32,38,37,40);(1,3,4,7)(2,5,6,8)(9,11)(10,14)(12,15)(13,16)(17,20)(18,19)(21,24)(22,23)(25,28,29,35)(26,32,30,37)(27,33,34,39)(31,38,36,40)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=h,d^2=fh,e^2=ghi,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, 5509357);