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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1508 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(11,16)(12,14)(17,18)(19,20)(21,23)(22,24)(25,26,30,29)(27,31,33,35)(28,32,38,37)(34,39,36,40);(1,2)(3,5)(4,6)(7,8)(9,10,13,15)(11,14,16,12)(26,29)(27,33)(32,37)(34,36);(1,2,4,6)(3,5,8,7)(9,11,13,16)(10,12,15,14)(17,19,22,21)(18,23,24,20)(25,27)(26,31)(28,36)(29,35)(30,33)(32,40)(34,38)(37,39);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,20,22,23)(18,19,24,21)(25,28,30,38)(26,32,29,37)(27,34,33,36)(31,39,35,40);(1,3,4,8)(2,5,6,7)(9,12,13,14)(10,11,15,16)(17,21,22,19)(18,23,24,20)(25,29,30,26)(27,35,33,31)(28,37,38,32)(34,40,36,39)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=g,d^2=gh,e^2=fg,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, 4859973);