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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1690 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)(3,5,7,8)(10,13)(14,16)(17,18)(19,23)(20,22)(21,24)(25,26)(27,30)(28,33)(29,32)(31,36)(34,37)(35,39)(38,40);(11,15)(14,16)(17,18)(19,23)(20,22)(21,24)(26,27)(31,33)(32,34)(38,39);(9,10,12,13)(11,14,15,16)(17,18,20,22)(19,21,24,23)(25,27,30,26)(28,31,36,33)(29,32,37,34)(35,39,40,38);(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,21)(20,22)(23,24)(25,28,30,36)(26,31,27,33)(29,35,37,40)(32,38,34,39);(1,3,4,7)(2,5,6,8)(9,11,12,15)(10,14,13,16)(17,19,20,24)(18,21,22,23)(25,29,30,37)(26,32,27,34)(28,35,36,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=fghi,c^2=g,d^2=f,e^2=fhi,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, 5497756);