Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| 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 | |
1632 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)(3,8)(4,7)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18,20,22)(19,23,24,21)(27,33)(28,35)(30,36)(31,38);(1,3,6,4)(2,7,5,8)(10,12)(11,15)(19,24)(21,23)(27,33)(30,36)(34,39)(37,40);(3,4)(7,8)(9,10,13,12)(11,14,15,16)(17,19,20,24)(18,21,22,23)(25,26,29,32)(27,30,33,36)(28,31,35,38)(34,37,39,40);(1,4,6,3)(2,7,5,8)(9,11,13,15)(10,14,12,16)(17,18)(19,23)(20,22)(21,24)(25,27)(26,30)(28,34)(29,33)(31,37)(32,36)(35,39)(38,40);(1,5,6,2)(3,7,4,8)(9,12,13,10)(11,16,15,14)(25,28)(26,31)(27,34)(29,35)(30,37)(32,38)(33,39)(36,40)Polycyclic presentation:
2^9|a^2=fh,b^2=gh,c^2=fghi,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5926624);