Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806623 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2067 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6)(7,10)(8,12)(9,11);(5,7,10,6)(8,12,9,11)(13,14)(15,24)(16,21)(17,30)(18,33)(19,23)(20,25)(22,31)(26,40)(27,34)(28,36)(29,37)(32,43)(35,41)(38,39)(42,44);(2,4)(13,15,18,25,19,27,16,28)(14,20,21,24,23,36,33,34)(17,26,31,39,32,41,29,42)(22,35,30,38,37,40,43,44);(5,8,10,9)(6,11,7,12)(13,16,19,18)(14,21,23,33)(15,25,27,28)(17,29,32,31)(20,34,36,24)(22,30,37,43)(26,39,41,42)(35,44,40,38);(5,9,10,8)(6,11,7,12)(13,17,16,29,19,32,18,31)(14,22,33,43,23,37,21,30)(15,26,28,42,27,41,25,39)(20,35,34,44,36,40,24,38)Polycyclic presentation:
2^9|b^2=f,c^2=h,d^2=fi,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6718555);