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 | |
1505 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,8)(4,6)(10,13)(14,15)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,36)(29,30)(32,38)(33,40)(34,35)(37,39);(1,2)(3,4)(5,7)(6,8)(17,19,22,20)(18,23,21,24)(26,29)(31,34)(32,36)(39,40);(1,3,5,8)(2,6,7,4)(9,10,12,13)(11,15,16,14)(17,20,22,19)(18,23,21,24)(25,27,30,35)(26,31,29,34)(28,37,38,33)(32,40,36,39);(1,3)(2,4)(5,8)(6,7)(9,11,12,16)(10,14,13,15)(17,21,22,18)(19,24,20,23)(25,28)(26,32)(27,33)(29,36)(30,38)(31,39)(34,40)(35,37);(1,4)(2,3)(5,6)(7,8)(25,29,30,26)(27,34,35,31)(28,36,38,32)(33,40,37,39)Polycyclic presentation:
2^9|b^2=fhi,c^2=fg,d^2=gh,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, 4860362);