Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| 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 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1255 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)(4,8)(5,6)(9,10,12,14)(11,15,16,13)(17,18,22,26)(19,29,31,24)(20,25,32,39)(21,35,36,27)(23,28,33,41)(30,38,45,44)(34,42,46,40)(37,48,47,43);(9,11)(10,13)(12,16)(14,15)(17,19,22,31)(18,24,26,29)(20,33,32,23)(21,34,36,46)(25,28,39,41)(27,42,35,40)(30,37,45,47)(38,43,44,48);(2,6)(7,8)(11,16)(13,15)(17,20,23,19,22,32,33,31)(18,25,28,29,26,39,41,24)(21,34,37,30,36,46,47,45)(27,40,43,44,35,42,48,38);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(18,26)(21,36)(24,29)(25,39)(28,41)(30,45)(34,46)(37,47);(1,4,5,3)(2,8,6,7)(17,21,33,47,22,36,23,37)(18,27,41,48,26,35,28,43)(19,30,20,34,31,45,32,46)(24,38,39,42,29,44,25,40)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fhi,d^2=g,e^2=fgh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,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, 6751287);