Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 | |
2665 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,4)(6,8)(9,10)(11,18)(12,13)(14,15)(16,21)(17,22)(19,24)(20,23)(26,30)(27,33)(32,35)(34,39);(10,15)(13,22)(16,18)(20,24)(25,26,29,30)(27,31,33,37)(28,35,36,32)(34,40,39,38);(5,6,7,8)(9,11,14,21)(10,16,15,18)(12,19,17,23)(13,20,22,24)(25,26)(27,31)(28,32)(29,30)(33,37)(34,38)(35,36)(39,40);(1,2)(3,4)(5,6)(7,8)(9,12,14,17)(10,13,15,22)(11,19,21,23)(16,20,18,24)(25,27,29,33)(26,31,30,37)(28,34,36,39)(32,38,35,40);(9,13)(10,17)(11,20)(12,15)(14,22)(16,23)(18,19)(21,24)(25,28)(26,32)(27,34)(29,36)(30,35)(31,38)(33,39)(37,40)Polycyclic presentation:
2^9|b^2=h,c^2=gh,d^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, 4841322);