Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3288 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C2×D42 |
| prime radical | C2×#3150 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2666 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,4,3)(5,6,11,16,12,17,10,15)(7,18,21,13,23,30,22,31)(8,24,25,33,26,32,9,14)(19,35,27,34,20,28,36,29);(1,3,4,2)(5,7)(6,13)(8,19)(9,27)(10,21)(11,22)(12,23)(14,28)(15,30)(16,18)(17,31)(20,26)(24,34)(25,36)(29,32)(33,35);(5,8,10,9,12,26,11,25)(6,14,16,24,17,33,15,32)(7,19,21,27,23,20,22,36)(13,28,18,34,31,35,30,29);(5,9,10,26,12,25,11,8)(6,14,15,32,17,33,16,24)(7,20,22,27,23,19,21,36)(13,29,18,28,31,34,30,35);(5,10,12,11)(6,15,17,16)(7,21,23,22)(8,9,26,25)(13,30,31,18)(14,32,33,24)(19,27,20,36)(28,29,35,34)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=g,d^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,d^b=df,f^b=fi,h^b=hi,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6290007);