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) = C26 |
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 | |
1584 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)(14,16)(17,18)(19,23)(20,21)(22,24)(25,26,30,31)(27,28,32,36)(29,33,37,38)(34,35,39,40);(1,3)(2,5)(4,8)(6,7)(10,13)(14,16)(17,19,22,20)(18,23,24,21)(26,31)(27,32)(29,37)(35,40);(2,4)(3,7)(9,10,12,13)(11,14,15,16)(17,20,22,19)(18,23,24,21)(25,27)(26,28)(29,34)(30,32)(31,36)(33,35)(37,39)(38,40);(1,4,6,2)(3,8,7,5)(17,21,22,23)(18,20,24,19)(25,28)(26,32)(27,31)(29,35)(30,36)(33,39)(34,38)(37,40);(1,5,6,8)(2,7,4,3)(9,11,12,15)(10,14,13,16)(25,29)(26,33)(27,34)(28,35)(30,37)(31,38)(32,39)(36,40)Polycyclic presentation:
2^9|a^2=f,b^2=gi,c^2=fgh,d^2=g,e^2=fh,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, 4864815);