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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1248 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(3,7)(9,10)(11,15)(12,16)(13,14)(17,18)(19,24)(20,30)(21,23)(22,26)(25,32)(27,39)(28,43)(29,35)(31,40)(33,45)(34,36)(37,42)(38,44)(41,47)(46,48);(1,2,4,5)(3,6,7,8)(18,23)(19,27)(20,31)(22,34)(24,37)(25,40)(29,42)(30,45)(32,33)(35,39)(38,43)(41,46);(10,14)(15,16)(17,19,22,27,21,29,34,42)(18,24,26,39,23,35,36,37)(20,28,33,41,32,44,31,46)(25,38,40,48,30,43,45,47);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,15,14,16)(18,23)(20,32)(24,35)(26,36)(28,44)(31,33)(37,39)(41,46);(2,5)(6,8)(9,12,13,11)(10,16,14,15)(17,20,22,33,21,32,34,31)(18,25,26,40,23,30,36,45)(19,28,27,41,29,44,42,46)(24,38,39,48,35,43,37,47)Polycyclic presentation:
2^9|b^2=fi,c^2=h,d^2=fgi,e^2=gh,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, 6748419);