Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6131635 |
| 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 | |
1466 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(3,7)(9,10)(11,13)(12,15)(14,16)(17,18,21,22)(19,20,24,23)(25,26)(27,28)(29,32)(30,38)(31,34)(33,39)(35,40)(36,37);(2,5)(6,8)(10,11)(14,15)(18,22)(20,23)(25,27,31,29)(26,32,34,28)(30,35,39,37)(33,40,38,36);(17,19)(18,23)(20,22)(21,24)(25,28,31,32)(26,27,34,29)(30,36,39,40)(33,35,38,37);(1,2)(3,6)(4,5)(7,8)(9,11,13,10)(12,15,16,14)(17,19)(18,23)(20,22)(21,24)(25,29,31,27)(26,32,34,28)(30,37,39,35)(33,40,38,36);(1,3)(2,6)(4,7)(5,8)(9,12,13,16)(10,14,11,15)(17,20,21,23)(18,19,22,24)(25,30,31,39)(26,33,34,38)(27,35,29,37)(28,36,32,40)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=eg68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6039339);