Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907869 |
| 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 | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1482 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)(17,18,23,27)(19,24,29,38)(20,32,35,46)(21,26,22,36)(25,33,40,34)(28,41,42,48)(30,39,31,45)(37,43,47,44);(1,3,7,4)(2,5,8,6)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,28)(21,30)(22,31)(23,29)(25,37)(26,39)(27,38)(32,41)(33,43)(34,44)(35,42)(36,45)(40,47)(46,48);(1,4,7,3)(2,5,8,6)(9,11)(10,13)(12,16)(14,15)(17,20,22,34,23,35,21,33)(18,25,26,32,27,40,36,46)(19,28,31,44,29,42,30,43)(24,37,39,41,38,47,45,48);(1,5,7,6)(2,3,8,4)(11,16)(13,15)(19,29)(24,38)(28,42)(30,31)(37,47)(39,45)(41,48)(43,44);(1,6,7,5)(2,3,8,4)(10,14)(13,15)(17,21,23,22)(18,26,27,36)(19,30,29,31)(20,33,35,34)(24,39,38,45)(25,32,40,46)(28,43,42,44)(37,41,47,48)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fg,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6822856);