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 | #45236 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1726 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)(17,18,23,24,22,28,21,27)(19,25,31,38,29,42,32,40)(20,33,36,44,37,26,34,39)(30,43,45,48,35,41,46,47);(1,3,4,7)(2,5,6,8)(9,10)(11,15)(12,14)(13,16)(18,24)(19,29)(20,34)(21,23)(25,40)(27,28)(30,45)(35,46)(36,37)(38,42)(39,44)(41,43);(2,6)(5,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,29)(18,25,28,42)(20,35,37,30)(21,32,23,31)(24,38,27,40)(26,43,33,41)(34,45,36,46)(39,48,44,47);(11,16)(13,15)(17,20,21,34,22,37,23,36)(18,26,27,44,28,33,24,39)(19,30,32,46,29,35,31,45)(25,41,40,48,42,43,38,47);(10,14)(13,15)(17,21,22,23)(18,24,28,27)(19,31,29,32)(20,36,37,34)(25,40,42,38)(26,44,33,39)(30,46,35,45)(41,47,43,48)Polycyclic presentation:
2^9|a^2=fgi,b^2=g,c^2=i,d^2=f,e^2=i,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, 6926488);