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, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1693 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,7,8,5)(10,11)(14,15)(17,18)(19,23)(20,21)(22,24)(27,33)(31,36)(34,39)(38,40);(3,8)(5,7)(9,10,13,11)(12,14,16,15)(18,21)(19,24)(26,30)(27,33)(32,37)(34,39);(10,11)(12,16)(17,19,20,24)(18,22,21,23)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(1,3)(2,5)(4,8)(6,7)(9,11,13,10)(12,15,16,14)(17,18)(19,22)(20,21)(23,24)(25,27)(26,31)(28,34)(29,33)(30,36)(32,38)(35,39)(37,40);(1,2)(3,5)(4,6)(7,8)(9,12,13,16)(10,14,11,15)(17,19)(18,22)(20,24)(21,23)(25,28,29,35)(26,32,30,37)(27,34,33,39)(31,38,36,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=fg,c^2=g,d^2=fg,e^2=gh,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, 6085663);