Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667484 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#126 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1257 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,17)(11,18,21,15)(12,16,13,22)(19,24,20,23)(25,26,29,31)(27,32,36,38)(28,33,35,39)(30,34,37,40);(9,11,14,21)(10,15,17,18)(12,19,13,20)(16,23,22,24)(26,31)(27,35)(28,36)(30,37)(32,33)(38,39);(2,5)(6,8)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,27,29,36)(26,32,31,38)(28,37,35,30)(33,40,39,34);(1,3,4,7)(2,6,5,8)(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(26,31)(32,38)(33,39)(34,40);(11,21)(15,18)(19,20)(23,24)(25,28,29,35)(26,33,31,39)(27,30,36,37)(32,34,38,40)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=f,d^2=fgi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,h^c=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6609707);