Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6349298 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #60790 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2671 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,7)(3,10)(4,5)(9,13)(11,12)(14,15)(17,18)(19,28)(20,25)(21,26)(22,36)(23,27)(24,31)(29,42)(30,38)(32,41)(33,39)(34,40)(35,46)(37,45)(43,48)(44,47);(1,2,6,9)(3,8,12,16)(4,7,5,13)(10,14,11,15)(17,19,23,32)(18,24,27,38)(20,29,33,43)(21,31,22,30)(25,37,39,47)(26,41,36,28)(34,45,35,44)(40,48,46,42);(1,3,6,12)(2,8,9,16)(4,10,5,11)(7,14,13,15)(17,20,23,33)(18,25,27,39)(19,29,32,43)(21,34,22,35)(24,37,38,47)(26,40,36,46)(28,42,41,48)(30,44,31,45);(1,4,6,5)(2,7,9,13)(3,11,12,10)(8,15,16,14)(20,33)(25,39)(29,43)(34,35)(37,47)(40,46)(42,48)(44,45);(2,9)(7,13)(8,16)(14,15)(17,21,23,22)(18,26,27,36)(19,30,32,31)(20,34,33,35)(24,28,38,41)(25,40,39,46)(29,44,43,45)(37,42,47,48)Polycyclic presentation:
2^9|b^2=h,c^2=h,d^2=i,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fh,g^a=gi,e^b=eh,f^b=fh,g^b=gi,d^c=dh,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6256562);