Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996405 |
| 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 | |
2211 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10,14,19,15,20,13,18)(11,21,23,33,24,32,30,16)(12,17,28,36,29,26,27,35)(22,34,25,37,38,39,40,31);(1,2)(3,7)(4,6)(5,8)(9,11,15,24)(10,16,20,33)(12,25,29,40)(13,23,14,30)(17,34,26,39)(18,21,19,32)(22,27,38,28)(31,35,37,36);(1,3)(2,5)(4,8)(6,7)(9,10,15,20)(11,16,24,33)(12,26,29,17)(13,18,14,19)(21,30,32,23)(22,37,38,31)(25,39,40,34)(27,36,28,35);(3,8)(5,7)(9,12,13,27,15,29,14,28)(10,17,18,35,20,26,19,36)(11,22,30,40,24,38,23,25)(16,31,32,39,33,37,21,34);(2,6)(5,7)(9,13,15,14)(10,18,20,19)(11,23,24,30)(12,27,29,28)(16,21,33,32)(17,35,26,36)(22,25,38,40)(31,34,37,39)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=fi,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=dhi,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6903610);