Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | |
1421 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)(3,8,4,7)(9,10,13,12)(11,15,16,14)(20,24)(22,23)(25,26)(27,33)(28,36)(29,30)(31,35)(32,37)(34,38)(39,40);(1,3,6,4)(2,7,5,8)(9,10)(11,15)(12,13)(14,16)(17,18,21,19)(20,22,24,23)(26,29)(31,33)(32,36)(38,39);(3,4)(7,8)(9,11)(10,14)(12,15)(13,16)(17,19,21,18)(20,23,24,22)(25,27,30,35)(26,31,29,33)(28,34,37,40)(32,38,36,39);(1,4,6,3)(2,7,5,8)(9,11)(10,15)(12,14)(13,16)(17,20)(18,22)(19,23)(21,24)(25,28,30,37)(26,32,29,36)(27,34,35,40)(31,38,33,39);(1,5,6,2)(3,8,4,7)(9,12,13,10)(11,15,16,14)(25,29,30,26)(27,33,35,31)(28,32,37,36)(34,38,40,39)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=i,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5666547);