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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1895 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)(11,12)(13,14)(15,16)(17,18,21,24)(19,20,23,22)(26,30)(27,33)(32,37)(35,39);(1,3,5,8)(2,6,4,7)(10,14)(11,16)(17,19,21,23)(18,22,24,20)(28,36)(32,37)(35,39)(38,40);(1,2,5,4)(3,7,8,6)(9,11)(10,15)(12,14)(13,16)(17,18)(19,22)(20,23)(21,24)(25,26,29,30)(27,34,33,31)(28,32,36,37)(35,40,39,38);(1,4,5,2)(3,7,8,6)(9,12,13,15)(10,11,14,16)(17,20,21,22)(18,23,24,19)(25,27,29,33)(26,31,30,34)(28,35,36,39)(32,38,37,40);(1,3)(2,7)(4,6)(5,8)(9,11)(10,12)(13,16)(14,15)(17,18)(19,20)(21,24)(22,23)(25,28,29,36)(26,32,30,37)(27,35,33,39)(31,38,34,40)Polycyclic presentation:
2^9|a^2=fh,b^2=ghi,c^2=i,d^2=g,e^2=fg,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, 5658868);