Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | |
1726 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)(3,6)(4,8)(5,7)(10,14)(12,16)(19,23)(22,24)(25,26,29,30)(27,33,35,31)(28,36,38,32)(34,39,37,40);(1,3,5,8)(2,6,7,4)(10,14)(11,15)(18,21)(22,24)(26,30)(31,33)(32,36)(39,40);(1,3)(2,4)(5,8)(6,7)(9,10,13,14)(11,12,15,16)(17,18)(19,22)(20,21)(23,24)(25,27,29,35)(26,31,30,33)(28,37,38,34)(32,40,36,39);(1,3)(2,4)(5,8)(6,7)(9,11,13,15)(10,12,14,16)(25,26)(27,31)(28,32)(29,30)(33,35)(34,39)(36,38)(37,40);(1,4)(2,3)(5,6)(7,8)(9,12)(10,15)(11,14)(13,16)(17,19)(18,22)(20,23)(21,24)(25,28,29,38)(26,32,30,36)(27,34,35,37)(31,39,33,40)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=fgi,d^2=gi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4621427);