Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470894 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #2619 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1428 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,31)(20,37)(21,27)(22,28)(23,29)(24,30)(25,34)(26,38)(32,42)(33,46)(35,45)(36,44)(39,43)(40,47)(41,48);(2,5)(6,8)(10,14)(11,12)(17,19,23,35,24,36,22,34)(18,25,28,44,30,45,29,31)(20,32,38,41,39,21,33,40)(26,42,37,47,46,27,43,48);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,15,14,16)(17,20,22,33,24,39,23,38)(18,26,29,43,30,46,28,37)(19,32,34,40,36,21,35,41)(25,42,31,48,45,27,44,47);(9,12,13,11)(10,16,14,15)(17,21,24,32)(18,27,30,42)(19,33,36,38)(20,34,39,35)(22,41,23,40)(25,43,45,37)(26,31,46,44)(28,48,29,47);(10,14)(15,16)(17,22,24,23)(18,28,30,29)(19,34,36,35)(20,38,39,33)(21,40,32,41)(25,44,45,31)(26,43,46,37)(27,47,42,48)Polycyclic presentation:
2^9|b^2=g,c^2=f,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,d^c=di,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6390235);