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 | C26 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2128 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(3,7)(10,13)(11,15)(19,23)(21,24)(25,26,30,28)(27,33,34,31)(29,36,37,32)(35,38,40,39);(2,5)(6,8)(17,18,20,22)(19,21,23,24)(26,28)(27,34)(29,37)(38,39);(9,10,12,13)(11,14,15,16)(17,18)(19,21)(20,22)(23,24)(25,27,30,34)(26,31,28,33)(29,35,37,40)(32,38,36,39);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(25,28,30,26)(27,33,34,31)(29,36,37,32)(35,39,40,38);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,15)(13,16)(17,19)(18,21)(20,23)(22,24)(25,29)(26,32)(27,35)(28,36)(30,37)(31,38)(33,39)(34,40)Polycyclic presentation:
2^9|a^2=f,b^2=fh,c^2=g,d^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, 5865828);