Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle, prime radical | C23 | C24×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1118 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,22)(20,27)(21,23)(24,28)(25,30)(26,32)(29,31)(33,34,39,40,38,42,35,43)(36,46,45,48,37,41,44,47);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19,21,25)(18,22,23,30)(20,26,28,31)(24,29,27,32)(33,35,38,39)(34,40,42,43)(36,44,37,45)(41,47,46,48);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(18,23)(19,25)(24,27)(26,31)(33,36,35,44,38,37,39,45)(34,41,43,48,42,46,40,47);(3,11)(6,9)(10,16)(13,15)(17,20,21,28)(18,24,23,27)(19,26,25,31)(22,29,30,32)(33,35,38,39)(34,40,42,43)(36,44,37,45)(41,47,46,48);(2,8)(6,9)(7,14)(13,15)(33,37,39,44,38,36,35,45)(34,41,40,47,42,46,43,48)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=fh,d^2=g,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cgi,d^b=dfi,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6338885);