Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7563333 |
| 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 | |
1656 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,16,21)(11,22,26,28)(12,27,20,15)(13,30,32,19)(14,18,33,29)(17,25,36,24)(23,34,40,35)(31,38,39,37);(1,3)(2,5)(4,7)(6,8)(9,11,15,24,16,26,27,25)(10,17,20,22,21,36,12,28)(13,31,14,23,32,39,33,40)(18,37,30,34,29,38,19,35);(9,12,15,10,16,20,27,21)(11,17,25,28,26,36,24,22)(13,19,14,18,32,30,33,29)(23,34,31,37,40,35,39,38);(3,7)(5,8)(9,13,27,33,16,32,15,14)(10,18,12,19,21,29,20,30)(11,23,24,39,26,40,25,31)(17,34,22,38,36,35,28,37);(9,14,15,32,16,33,27,13)(10,19,20,18,21,30,12,29)(11,23,24,39,26,40,25,31)(17,35,22,37,36,34,28,38)Polycyclic presentation:
2^9|a^2=gh,b^2=f,c^2=g,d^2=gi,e^2=f,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7500217);