Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7021363 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2129 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10,14,19,15,20,31,18)(11,21,23,35,25,34,24,16)(12,26,13,17,29,37,28,38)(22,36,30,40,27,32,39,33);(1,2)(3,7)(4,6)(5,8)(9,11)(10,16)(12,27)(13,30)(14,24)(15,25)(17,36)(18,21)(19,34)(20,35)(22,29)(23,31)(26,40)(28,39)(32,38)(33,37);(1,3)(2,5)(4,8)(6,7)(9,12,14,13,15,29,31,28)(10,17,18,26,20,38,19,37)(11,22,24,39,25,27,23,30)(16,32,21,33,35,36,34,40);(3,8)(5,7)(9,13,14,29,15,28,31,12)(10,17,19,37,20,38,18,26)(11,22,23,30,25,27,24,39)(16,33,21,36,35,40,34,32);(2,6)(5,7)(9,14,15,31)(10,18,20,19)(11,23,25,24)(12,28,29,13)(16,34,35,21)(17,37,38,26)(22,39,27,30)(32,33,36,40)Polycyclic presentation:
2^9|a^2=f,c^2=g,d^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6928565);