Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573511 |
| 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 | #33187 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1616 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,6)(3,7,4,8)(9,10)(11,14)(12,13)(15,16)(17,18,22,27)(19,24,31,41)(20,32,34,25)(21,26,37,45)(23,28,30,40)(29,47,48,39)(33,44,36,42)(35,46,38,43);(2,6)(7,8)(17,19,22,31)(18,24,27,41)(20,33,34,36)(21,23,37,30)(25,42,32,44)(26,28,45,40)(29,38,48,35)(39,46,47,43);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,22,34)(18,25,27,32)(19,29,31,48)(21,36,37,33)(23,38,30,35)(24,39,41,47)(26,44,45,42)(28,46,40,43);(1,3,5,4)(2,7,6,8)(9,11,12,15)(10,14,13,16)(17,21,22,37)(18,26,27,45)(19,30,31,23)(20,29,34,48)(24,40,41,28)(25,39,32,47)(33,35,36,38)(42,43,44,46);(1,4,5,3)(2,8,6,7)(19,21)(23,30)(24,26)(28,40)(29,36)(31,37)(33,48)(35,38)(39,44)(41,45)(42,47)(43,46)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=fi,d^2=fg,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6496907);