Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519075 |
| 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 | #3498 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1737 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)(4,8)(5,6)(9,10)(11,21)(12,26)(13,14)(15,20)(16,19)(17,25)(18,30)(22,34)(23,31)(24,33)(27,32)(28,36)(29,35)(37,38)(39,40);(2,6)(7,8)(9,11,15,25)(10,17,20,21)(12,27,29,38)(13,23,19,33)(14,24,16,31)(18,34,36,40)(22,30,39,28)(26,37,35,32);(1,3,5,4)(2,7,6,8)(9,12,14,28,15,29,16,30)(10,18,19,35,20,36,13,26)(11,22,31,27,25,39,24,38)(17,32,23,34,21,37,33,40);(1,3,5,4)(2,7,6,8)(9,13,14,10,15,19,16,20)(11,23,31,21,25,33,24,17)(12,26,28,18,29,35,30,36)(22,34,27,37,39,40,38,32);(1,4,5,3)(2,7,6,8)(9,14,15,16)(10,19,20,13)(11,24,25,31)(12,28,29,30)(17,33,21,23)(18,35,36,26)(22,38,39,27)(32,40,37,34)Polycyclic presentation:
2^9|b^2=i,c^2=fg,d^2=fg,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7472260);