Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996375 |
| 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 | #2011 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2402 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1325 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,7)(3,8,4,6)(9,10)(11,13)(12,14)(15,16)(17,18,23,24,22,27,21,28)(19,25,34,36,30,40,31,39)(20,26,35,37,33,41,32,42)(29,38,45,46,44,48,43,47);(9,10)(11,15)(12,14)(13,16)(18,24)(20,32)(21,23)(25,36)(27,28)(29,43)(31,34)(33,35)(37,42)(39,40)(44,45)(46,47);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,19,22,30)(18,25,27,40)(20,29,33,44)(21,31,23,34)(24,36,28,39)(26,38,41,48)(32,43,35,45)(37,46,42,47);(11,16)(13,15)(17,20,21,32,22,33,23,35)(18,26,28,42,27,41,24,37)(19,29,31,43,30,44,34,45)(25,38,39,47,40,48,36,46);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(18,27)(19,30)(24,28)(26,41)(29,44)(31,34)(37,42)(43,45)Polycyclic presentation:
2^9|a^2=fgi,c^2=ghi,d^2=f,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6999599);