Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2601293 |
| 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 | |
1430 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,7,8,5)(11,15)(12,14)(17,18,21,19)(20,22,24,23)(25,26,29,30)(27,33,34,31)(28,36,38,32)(35,39,37,40);(1,2,4,6)(3,5,8,7)(9,10,13,16)(11,12,15,14)(17,19,21,18)(20,22,24,23)(26,30)(27,34)(32,36)(35,37);(1,2)(3,5)(4,6)(7,8)(9,11,13,15)(10,14,16,12)(17,20)(18,22)(19,23)(21,24)(25,26)(27,31)(28,32)(29,30)(33,34)(35,39)(36,38)(37,40);(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,15)(13,16)(25,27,29,34)(26,31,30,33)(28,37,38,35)(32,40,36,39);(1,3,4,8)(2,5,6,7)(9,12,13,14)(10,15,16,11)(25,28)(26,32)(27,35)(29,38)(30,36)(31,39)(33,40)(34,37)Polycyclic presentation:
2^9|a^2=ghi,b^2=fh,c^2=gi,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2541415);