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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1554 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)(9,10,12,13)(11,15,16,14)(17,18,22,23)(19,24,21,20)(25,26)(27,32)(28,29)(30,33)(31,34)(35,39)(36,37)(38,40);(3,8)(5,7)(10,13)(14,15)(17,19)(18,20)(21,22)(23,24)(25,27,31,28)(26,32,34,29)(30,36,38,35)(33,37,40,39);(10,13)(14,15)(18,23)(19,21)(25,28,31,27)(26,32,34,29)(30,36,38,35)(33,39,40,37);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,16)(13,15)(17,20,22,24)(18,19,23,21)(25,29,31,32)(26,28,34,27)(30,37,38,39)(33,36,40,35);(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,21,22,19)(18,24,23,20)(25,30)(26,33)(27,35)(28,36)(29,37)(31,38)(32,39)(34,40)Polycyclic presentation:
2^9|a^2=hi,b^2=fg,c^2=fg,d^2=fgh,e^2=h,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, 6092106);