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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2401 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1360 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,13)(12,14)(15,16)(17,18,24,28,23,30,22,29)(19,31,37,25,34,44,33,45)(20,35,38,46,40,26,39,43)(21,41,36,47,42,27,32,48);(9,10)(11,15)(12,14)(13,16)(17,19,23,34)(18,25,30,45)(20,36,40,32)(21,39,42,38)(22,37,24,33)(26,41,35,27)(28,31,29,44)(43,48,46,47);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,20,23,40)(18,26,30,35)(19,32,34,36)(21,33,42,37)(22,39,24,38)(25,27,45,41)(28,43,29,46)(31,47,44,48);(11,16)(13,15)(17,21,24,36,23,42,22,32)(18,27,28,48,30,41,29,47)(19,20,37,38,34,40,33,39)(25,43,44,35,45,46,31,26);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,22,23,24)(18,28,30,29)(19,33,34,37)(20,38,40,39)(21,36,42,32)(25,44,45,31)(26,46,35,43)(27,47,41,48)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=h,d^2=fi,e^2=g,f^2=i,b^a=bf,c^a=cg,d^a=di,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, 6993605);