Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907993 |
| 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 | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1784 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)(17,18,22,27)(19,28,23,24)(20,31,33,39)(21,35,37,26)(25,29,42,34)(30,40,38,44)(32,48,36,47)(41,46,43,45);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(17,19,22,23)(18,24,27,28)(20,29,33,34)(21,30,37,38)(25,39,42,31)(26,40,35,44)(32,45,36,46)(41,48,43,47);(3,7)(6,8)(9,11)(10,13)(12,16)(14,15)(17,20,19,29,22,33,23,34)(18,25,24,39,27,42,28,31)(21,36,30,46,37,32,38,45)(26,43,40,47,35,41,44,48);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(17,21)(18,26)(19,30)(20,32)(22,37)(23,38)(24,40)(25,41)(27,35)(28,44)(29,45)(31,47)(33,36)(34,46)(39,48)(42,43);(2,5)(6,8)(10,14)(13,15)(21,37)(26,35)(30,38)(32,36)(40,44)(41,43)(45,46)(47,48)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=gi,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,d^c=dfh,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6816867);