Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
1458 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,7,8,5)(19,23)(21,24)(25,26)(27,31)(28,32)(29,30)(33,38)(34,39)(35,40)(36,37);(1,2)(3,5)(4,6)(7,8)(10,13)(14,16)(17,18,20,22)(19,21,23,24)(26,30)(27,33)(32,36)(34,35);(1,2)(3,7)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,21)(20,22)(23,24)(25,27,29,33)(26,31,30,38)(28,35,37,34)(32,40,36,39);(1,2)(3,7)(4,6)(5,8)(9,11,12,15)(10,14,13,16)(17,19)(18,21)(20,23)(22,24)(25,26)(27,31)(28,36)(29,30)(32,37)(33,38)(34,40)(35,39);(1,3,4,8)(2,5,6,7)(25,28,29,37)(26,32,30,36)(27,34,33,35)(31,39,38,40)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=fg,d^2=fhi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,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, 3408485);