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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1434 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,6)(3,7)(10,13)(14,16)(17,18,20,23)(19,22,24,21)(26,30)(27,33)(28,36)(38,40);(1,2,5,6)(3,4,7,8)(11,15)(14,16)(17,19,20,24)(18,21,23,22)(26,30)(28,36)(31,34)(35,39);(2,6)(3,7)(9,10,12,13)(11,14,15,16)(17,18)(19,22)(20,23)(21,24)(25,26)(27,34)(28,32)(29,30)(31,33)(35,40)(36,37)(38,39);(1,3)(2,4)(5,7)(6,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,21)(20,23)(22,24)(25,27,29,33)(26,31,30,34)(28,35,36,39)(32,38,37,40);(1,4)(2,7)(3,6)(5,8)(9,11)(10,14)(12,15)(13,16)(17,19)(18,22)(20,24)(21,23)(25,28,29,36)(26,32,30,37)(27,35,33,39)(31,38,34,40)Polycyclic presentation:
2^9|a^2=fgi,b^2=f,c^2=hi,d^2=fh,e^2=fh,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, 5650910);