Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519063 |
| 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 | |
1714 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)(9,10,15,20)(11,16,22,17)(12,23,25,33)(13,26,29,40)(14,30,31,19)(18,35,36,28)(21,37,27,38)(24,32,39,34);(2,6)(7,8)(9,11,15,22)(10,16,20,17)(12,14,25,31)(13,27,29,21)(18,32,36,34)(19,23,30,33)(24,35,39,28)(26,38,40,37);(1,3,5,4)(2,7,6,8)(9,12,14,22,15,25,31,11)(10,17,19,23,20,16,30,33)(13,24,28,27,29,39,35,21)(18,34,26,38,36,32,40,37);(9,13,14,28,15,29,31,35)(10,18,19,26,20,36,30,40)(11,21,12,24,22,27,25,39)(16,32,33,37,17,34,23,38);(1,4,5,3)(2,7,6,8)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fg,d^2=g,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=cg,d^b=dh,g^b=gi,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7471017);