Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #45092 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1733 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,27,23,28,21,24)(19,29,32,38,30,25,36,40)(20,26,33,43,35,44,37,39)(31,42,46,47,34,41,45,48);(1,3,4,7)(2,5,6,8)(9,10)(11,15)(12,14)(13,16)(18,24)(19,30)(20,33)(21,22)(25,40)(27,28)(29,38)(31,45)(34,46)(35,37)(39,43)(41,42);(2,6)(5,8)(9,11)(10,13)(12,16)(14,15)(17,19,23,30)(18,25,28,29)(20,34,35,31)(21,36,22,32)(24,38,27,40)(26,42,44,41)(33,45,37,46)(39,48,43,47);(11,16)(13,15)(17,20,22,33,23,35,21,37)(18,26,27,43,28,44,24,39)(19,31,32,46,30,34,36,45)(25,41,40,48,29,42,38,47);(10,14)(13,15)(17,21,23,22)(18,27,28,24)(19,32,30,36)(20,33,35,37)(25,38,29,40)(26,39,44,43)(31,45,34,46)(41,48,42,47)Polycyclic presentation:
2^9|a^2=fgi,b^2=gi,c^2=i,d^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6926464);