Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| 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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1790 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)(3,5)(4,6)(7,8)(11,15)(13,16)(19,23)(21,24)(25,26)(27,28)(29,34)(30,31)(32,33)(35,36)(37,40)(38,39);(3,7)(5,8)(9,10,12,14)(11,13,15,16)(17,18,20,22)(19,21,23,24)(26,31)(27,33)(34,39)(35,37);(1,2,4,6)(3,5,7,8)(9,11,12,15)(10,13,14,16)(17,18)(19,21)(20,22)(23,24)(25,27)(26,32)(28,31)(29,37)(30,33)(34,36)(35,38)(39,40);(1,2)(3,5)(4,6)(7,8)(9,11)(10,13)(12,15)(14,16)(25,28,30,32)(26,33,31,27)(29,36,38,40)(34,37,39,35);(1,3)(2,5)(4,7)(6,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,21)(20,23)(22,24)(25,29)(26,34)(27,35)(28,36)(30,38)(31,39)(32,40)(33,37)Polycyclic presentation:
2^9|b^2=hi,c^2=fg,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5389621);