Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6132643 |
| 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 | |
1372 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,3)(6,7)(9,10,14,13)(11,16,12,15)(17,18,20,22)(19,21,23,24)(25,26,30,33)(27,29,34,31)(28,32,38,40)(35,37,36,39);(1,2,5,3)(4,6,8,7)(9,10)(11,15)(12,16)(13,14)(17,19,20,23)(18,21,22,24)(27,34)(29,31)(35,36)(37,39);(2,3)(4,8)(9,11,14,12)(10,15,13,16)(17,19)(18,21)(20,23)(22,24)(25,27)(26,31)(28,36)(29,33)(30,34)(32,37)(35,38)(39,40);(1,3,5,2)(4,7,8,6)(9,12,14,11)(10,15,13,16)(25,28)(26,32)(27,35)(29,37)(30,38)(31,39)(33,40)(34,36);(1,4)(2,6)(3,7)(5,8)(9,13,14,10)(11,15,12,16)(17,18)(19,21)(20,22)(23,24)(25,29,30,31)(26,27,33,34)(28,37,38,39)(32,35,40,36)Polycyclic presentation:
2^9|a^2=fh,b^2=fghi,c^2=fi,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dfi,e^b=eg,d^c=dfg,e^c=ef62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6176720);