Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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×#466 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1277 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,6)(7,8)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26,31,36)(27,32,38,40)(28,34,37,39)(29,35,30,33);(1,2,5,6)(3,8,4,7)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27)(26,32)(28,29)(30,37)(31,38)(33,39)(34,35)(36,40);(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(26,33)(28,37)(29,30)(32,39)(34,40)(35,36);(1,3,5,4)(2,7,6,8)(12,20)(15,24)(17,19)(21,23)(25,28,31,37)(26,34,36,39)(27,29,38,30)(32,35,40,33);(1,4,5,3)(2,7,6,8)(10,16)(11,18)(15,24)(17,19)(25,29,31,30)(26,35,36,33)(27,37,38,28)(32,39,40,34)Polycyclic presentation:
2^9|a^2=hi,b^2=f,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,e^c=ei,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6947442);