Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 | |
3238 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,3,4)(5,6,7,8)(9,10)(11,15)(12,20)(13,14)(16,22)(17,24)(18,19)(21,23)(26,30)(27,33)(32,35)(34,39);(10,13)(15,18)(16,20)(23,24)(25,26,29,30)(27,31,33,37)(28,35,36,32)(34,40,39,38);(6,8)(9,11,14,19)(10,15,13,18)(12,21,22,17)(16,24,20,23)(25,26)(27,31)(28,32)(29,30)(33,37)(34,38)(35,36)(39,40);(1,2)(3,4)(5,6)(7,8)(9,12)(10,16)(11,17)(13,20)(14,22)(15,23)(18,24)(19,21)(25,27)(26,31)(28,34)(29,33)(30,37)(32,38)(35,40)(36,39);(9,13,14,10)(11,18,19,15)(12,20,22,16)(17,24,21,23)(25,28,29,36)(26,32,30,35)(27,34,33,39)(31,38,37,40)Polycyclic presentation:
2^9|a^2=fi,b^2=h,c^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4849426);