Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573523 |
| 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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1564 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)(19,24)(20,25)(21,35)(22,27)(23,28)(26,37)(29,39)(30,45)(31,40)(32,41)(33,43)(34,44)(36,48)(38,46)(42,47);(2,6)(7,8)(17,19,23,31)(18,24,28,40)(20,32,34,36)(21,22,37,30)(25,41,44,48)(26,45,35,27)(29,38,47,33)(39,46,42,43);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,23,34)(18,25,28,44)(19,29,31,47)(21,36,37,32)(22,38,30,33)(24,39,40,42)(26,41,35,48)(27,46,45,43);(1,3,5,4)(2,7,6,8)(9,11,12,15)(10,14,13,16)(17,21,23,37)(18,26,28,35)(19,30,31,22)(20,29,34,47)(24,27,40,45)(25,42,44,39)(32,33,36,38)(41,46,48,43);(1,4,5,3)(2,8,6,7)(19,21)(22,30)(24,35)(26,40)(27,45)(29,36)(31,37)(32,47)(33,38)(39,48)(41,42)(43,46)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fi,d^2=fgi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=di,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=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6496930);