Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518944 |
| 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 | C22×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1372 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)(3,7)(4,5)(6,8)(9,10,13,14)(11,16,12,15)(17,18,23,29)(19,27,33,25)(20,34,37,44)(21,39,32,41)(22,30,24,28)(26,36,45,38)(31,42,47,48)(35,40,46,43);(2,5)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19,23,33)(18,25,29,27)(20,35,37,46)(21,22,32,24)(26,42,45,48)(28,41,30,39)(31,38,47,36)(34,43,44,40);(10,14)(15,16)(17,20,22,36,23,37,24,38)(18,26,28,44,29,45,30,34)(19,31,21,35,33,47,32,46)(25,40,39,42,27,43,41,48);(9,11,13,12)(10,15,14,16)(17,21,24,19,23,32,22,33)(18,27,30,39,29,25,28,41)(20,35,38,31,37,46,36,47)(26,43,34,42,45,40,44,48);(1,3,4,8)(2,6,5,7)(9,12,13,11)(10,16,14,15)(17,22,23,24)(18,28,29,30)(19,32,33,21)(20,36,37,38)(25,41,27,39)(26,44,45,34)(31,46,47,35)(40,48,43,42)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=hi,d^2=g,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7511093);