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 |
| prime radical | #61080 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1965 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)(12,14)(13,20)(15,31)(16,22)(17,27)(18,32)(19,34)(21,35)(24,26)(25,36)(28,38)(29,37)(30,33)(39,40);(1,2)(3,6)(4,5)(7,8)(9,11,15,27,16,29,35,28)(10,17,21,23,22,38,31,37)(12,30,32,24,19,40,20,36)(13,33,34,25,18,39,14,26);(2,5)(6,7)(9,12,15,32,16,19,35,20)(10,18,21,14,22,13,31,34)(11,24,28,30,29,36,27,40)(17,26,37,39,38,25,23,33);(1,3,4,8)(2,6,5,7)(9,13,15,34,16,18,35,14)(10,19,21,20,22,12,31,32)(11,25,28,33,29,26,27,39)(17,36,37,40,38,24,23,30);(9,14,15,13,16,34,35,18)(10,20,21,12,22,32,31,19)(11,26,27,33,29,25,28,39)(17,36,23,30,38,24,37,40)Polycyclic presentation:
2^9|b^2=f,c^2=h,d^2=g,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^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7504249);