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 | C2×#553 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1465 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)(3,7,8,5)(9,10)(11,13)(12,14)(15,16)(17,18,23,24,22,28,21,27)(19,29,36,41,31,25,32,38)(20,33,37,42,35,26,34,39)(30,40,46,47,45,43,44,48);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(18,24)(20,34)(21,23)(25,38)(27,28)(29,41)(30,44)(32,36)(35,37)(39,42)(45,46)(47,48);(2,6)(5,7)(9,11,12,16)(10,13,14,15)(17,19,22,31)(18,25,28,29)(20,30,35,45)(21,32,23,36)(24,38,27,41)(26,40,33,43)(34,44,37,46)(39,47,42,48);(3,8)(5,7)(11,16)(13,15)(17,20,21,34,22,35,23,37)(18,26,27,42,28,33,24,39)(19,30,32,44,31,45,36,46)(25,40,41,48,29,43,38,47);(10,14)(13,15)(19,31)(20,35)(25,29)(26,33)(32,36)(34,37)(38,41)(39,42)Polycyclic presentation:
2^9|a^2=fi,c^2=hi,d^2=fgi,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, 6986929);