Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7563327 |
| 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 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1985 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)(3,7)(4,5)(6,8)(9,10)(11,23)(12,15)(13,20)(14,33)(16,21)(17,22)(18,25)(19,31)(24,34)(26,36)(27,28)(29,32)(30,38)(35,40)(37,39);(1,2)(3,6)(4,5)(7,8)(9,11,15,25,17,27,16,26)(10,18,21,23,22,36,12,28)(13,30,31,40,32,39,14,24)(19,37,20,35,33,38,29,34);(2,5)(6,7)(9,12,16,22,17,21,15,10)(11,18,25,23,27,36,26,28)(13,20,14,19,32,29,31,33)(24,35,30,38,40,34,39,37);(1,3,4,8)(2,6,5,7)(9,13,16,14,17,32,15,31)(10,19,12,29,22,33,21,20)(11,24,25,30,27,40,26,39)(18,34,23,37,36,35,28,38);(9,14,15,13,17,31,16,32)(10,20,21,33,22,29,12,19)(11,24,25,30,27,40,26,39)(18,35,23,38,36,34,28,37)Polycyclic presentation:
2^9|b^2=f,c^2=h,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,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7504232);