Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470037 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2872 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(6,8)(9,10,15,19,16,20,14,18)(11,21,23,28,24,29,30,13)(12,17,26,32,27,33,31,34)(22,36,38,40,39,35,37,25);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,22)(14,23)(15,30)(16,24)(17,25)(18,21)(19,29)(20,28)(26,37)(27,39)(31,38)(32,35)(33,40)(34,36);(2,5)(6,8)(9,12,16,27)(10,17,20,33)(11,22,24,39)(13,25,28,40)(14,26,15,31)(18,32,19,34)(21,35,29,36)(23,37,30,38);(2,5)(6,8)(9,13)(10,11)(12,25)(14,29)(15,21)(16,28)(17,22)(18,30)(19,23)(20,24)(26,36)(27,40)(31,35)(32,38)(33,39)(34,37);(1,3,4,7)(2,6,5,8)(9,14,16,15)(10,18,20,19)(11,23,24,30)(12,26,27,31)(13,21,28,29)(17,32,33,34)(22,37,39,38)(25,35,40,36)Polycyclic presentation:
2^9|a^2=f,c^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6398668);