Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667604 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1518 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,6)(5,8)(9,10,12,14)(11,15,16,13)(19,29)(21,34)(24,38)(26,40)(30,42)(31,37)(33,36)(35,41);(1,3)(2,5)(4,8)(6,7)(17,18,22,27)(19,30,29,42)(20,28,23,25)(21,35,34,41)(24,37,38,31)(26,36,40,33)(32,39,43,47)(44,45,48,46);(9,11,12,16)(10,13,14,15)(17,19,22,29)(18,24,27,38)(20,31,23,37)(21,32,34,43)(25,30,28,42)(26,39,40,47)(33,45,36,46)(35,44,41,48);(2,6)(5,7)(17,20,22,23)(18,25,27,28)(19,31,29,37)(21,36,34,33)(24,30,38,42)(26,41,40,35)(32,46,43,45)(39,48,47,44);(3,8)(5,7)(17,21,22,34)(18,26,27,40)(19,32,29,43)(20,33,23,36)(24,39,38,47)(25,35,28,41)(30,44,42,48)(31,45,37,46)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=g,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6595229);