Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1491 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)(3,7)(4,5)(6,8)(9,10,13,15)(11,14,16,12)(17,18,20,23)(19,21,24,22)(25,26,29,30)(27,33,34,31)(28,32,36,37)(35,39,40,38);(1,3,5,8)(2,6,4,7)(9,11,13,16)(10,12,15,14)(17,19)(18,21)(20,24)(22,23)(26,30)(27,34)(28,36)(38,39);(2,4)(6,7)(11,16)(12,14)(17,19,20,24)(18,22,23,21)(25,26)(27,31)(28,37)(29,30)(32,36)(33,34)(35,39)(38,40);(1,3)(2,7)(4,6)(5,8)(9,11)(10,14)(12,15)(13,16)(17,19)(18,22)(20,24)(21,23)(25,27,29,34)(26,31,30,33)(28,35,36,40)(32,38,37,39);(1,4,5,2)(3,6,8,7)(9,12,13,14)(10,11,15,16)(17,18)(19,22)(20,23)(21,24)(25,28)(26,32)(27,35)(29,36)(30,37)(31,38)(33,39)(34,40)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=fghi,d^2=fgh,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5184295);