Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1782 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,5)(3,7,8,6)(10,12)(11,15)(19,23)(22,24)(25,26)(27,31)(28,32)(29,30)(33,39)(34,38)(35,37)(36,40);(2,5)(6,7)(9,10,13,12)(11,16,15,14)(18,21)(22,24)(26,30)(31,38)(32,35)(39,40);(2,5)(3,8)(9,11,13,15)(10,14,12,16)(17,18,20,21)(19,22,23,24)(25,26)(27,31)(28,35)(29,30)(32,37)(33,40)(34,38)(36,39);(1,3)(2,6)(4,8)(5,7)(9,12,13,10)(11,14,15,16)(17,19,20,23)(18,22,21,24)(25,27,29,34)(26,31,30,38)(28,36,37,33)(32,40,35,39);(1,2)(3,7)(4,5)(6,8)(9,11)(10,14)(12,16)(13,15)(25,28)(26,32)(27,33)(29,37)(30,35)(31,39)(34,36)(38,40)Polycyclic presentation:
2^9|a^2=fghi,b^2=fhi,c^2=fgh,d^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4105760);