Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574070 |
| 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 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2577 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2)(3,4)(5,6)(7,11)(8,17)(9,12)(10,13)(14,18)(15,19)(16,20);(5,6,9,17,10,13,8,12)(7,11,18,19,16,20,15,14);(2,4)(5,6)(7,14)(8,12)(9,17)(10,13)(11,18)(15,20)(16,19);(5,7,10,16)(6,11,13,20)(8,15,9,18)(12,14,17,19);(5,6)(7,14)(8,12)(9,17)(10,13)(11,18)(15,20)(16,19)Polycyclic presentation:
2^9|b^2=fi,d^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6481270);