Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7563319 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 | |
2640 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,3)(5,6,11,12)(7,15,14,10)(8,17,18,13)(9,19,20,16);(2,3)(5,7,10,6,11,14,15,12)(8,17,9,16,18,13,20,19);(1,2,4,3)(6,12)(7,14)(13,17)(16,19);(1,3,4,2)(5,8,10,9,11,18,15,20)(6,13,7,16,12,17,14,19);(5,9,15,8,11,20,10,18)(6,13,7,16,12,17,14,19)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gi,d^2=h,e^2=fi,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,h^b=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7521288);