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×#449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2401 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1408 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)(17,18,21,27,22,26,33,23)(19,24,30,38,29,39,44,35)(20,31,32,36,34,25,40,41)(28,42,43,46,45,37,47,48);(9,10)(11,15)(12,14)(13,16)(18,23)(20,32)(21,33)(24,35)(26,27)(28,43)(30,44)(34,40)(36,41)(38,39)(45,47)(46,48);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,19,22,29)(18,24,26,39)(20,28,34,45)(21,30,33,44)(23,35,27,38)(25,37,31,42)(32,43,40,47)(36,46,41,48);(11,16)(13,15)(17,20,33,40,22,34,21,32)(18,25,23,36,26,31,27,41)(19,28,44,47,29,45,30,43)(24,37,35,46,39,42,38,48);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(18,26)(19,29)(20,34)(23,27)(30,44)(32,40)(37,42)(46,48)Polycyclic presentation:
2^9|a^2=fg,c^2=hi,d^2=fi,e^2=g,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,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, 6993729);