Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2601293 |
| 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 | |
1544 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)(3,5,8,7)(9,10,13,14)(11,16,15,12)(17,18,21,19)(20,23,24,22)(25,26)(27,33)(28,32)(29,30)(31,35)(34,40)(36,39)(37,38);(1,2)(3,7)(4,6)(5,8)(10,14)(11,15)(17,19,21,18)(20,23,24,22)(26,30)(31,33)(32,38)(39,40);(1,2)(3,5)(4,6)(7,8)(9,11,13,15)(10,12,14,16)(25,26)(27,31)(28,32)(29,30)(33,35)(34,39)(36,40)(37,38);(1,3,4,8)(2,5,6,7)(9,10)(11,12)(13,14)(15,16)(17,20,21,24)(18,22,19,23)(25,27,29,35)(26,31,30,33)(28,36,37,34)(32,40,38,39);(1,2)(3,7)(4,6)(5,8)(9,12)(10,15)(11,14)(13,16)(25,28,29,37)(26,32,30,38)(27,34,35,36)(31,39,33,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=fgh,d^2=gi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2566693);