Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 | |
1341 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,5)(4,6)(7,8)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22)(25,26,30,34)(27,35,39,51)(28,32,41,48)(29,43,36,33)(31,44,46,38)(37,50,40,55)(42,56,52,49)(45,53,47,54);(1,3,4,7)(2,5,6,8)(11,17)(12,20)(14,21)(15,19)(25,27,36,44,30,39,29,38)(26,31,43,51,34,46,33,35)(28,40,52,54,41,37,42,53)(32,47,56,50,48,45,49,55);(2,6)(5,8)(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,28,30,41)(26,32,34,48)(27,37,39,40)(29,42,36,52)(31,45,46,47)(33,49,43,56)(35,50,51,55)(38,54,44,53);(11,17)(14,21)(18,23)(22,24)(27,38)(28,41)(29,36)(31,35)(32,48)(33,43)(37,53)(39,44)(40,54)(45,55)(46,51)(47,50);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(27,39)(28,41)(31,46)(32,48)(35,51)(38,44)(42,52)(49,56)Polycyclic presentation:
2^9|a^2=h,b^2=fgi,c^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,e^c=ei,f^d=fi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7080412);