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 | |
1451 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,6,4)(3,8,5,7)(9,10)(11,13)(12,14)(15,16)(17,18,21,19)(20,23,24,22)(25,26,29,30)(27,31,35,34)(28,36,37,32)(33,39,40,38);(1,2,6,4)(3,7,5,8)(10,11)(14,15)(17,19,21,18)(20,23,24,22)(26,30)(31,34)(32,36)(38,39);(1,3,6,5)(2,7,4,8)(9,11,13,10)(12,14,16,15)(25,27)(26,31)(28,33)(29,35)(30,34)(32,38)(36,39)(37,40);(1,4,6,2)(3,7,5,8)(9,12)(10,14)(11,15)(13,16)(17,20)(18,22)(19,23)(21,24)(25,28,29,37)(26,32,30,36)(27,33,35,40)(31,38,34,39);(1,5,6,3)(2,7,4,8)(9,12)(10,14)(11,15)(13,16)(25,26)(27,34)(28,32)(29,30)(31,35)(33,39)(36,37)(38,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=fgi,c^2=g,d^2=fg,e^2=gh,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, 5397709);