Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | |
1626 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(5,8)(9,10,13,15)(11,12,16,14)(19,23)(22,24)(25,26)(27,32)(28,34)(29,33)(30,31)(35,37)(36,40)(38,39);(1,2,4,6)(3,5,7,8)(9,10,13,15)(11,12,16,14)(18,21)(22,24)(26,31)(27,34)(33,39)(35,40);(1,3,4,7)(2,5,6,8)(9,11,13,16)(10,14,15,12)(17,18,20,21)(19,22,23,24)(25,27,30,34)(26,32,31,28)(29,35,38,40)(33,37,39,36);(1,3)(2,5)(4,7)(6,8)(9,11)(10,12)(13,16)(14,15)(25,28,30,32)(26,27,31,34)(29,37,38,36)(33,40,39,35);(1,2)(3,5)(4,6)(7,8)(9,12)(10,11)(13,14)(15,16)(17,19)(18,22)(20,23)(21,24)(25,29,30,38)(26,33,31,39)(27,35,34,40)(28,36,32,37)Polycyclic presentation:
2^9|a^2=fgi,b^2=g,c^2=fhi,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5626400);