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 | |
1487 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)(17,18,22,25)(19,28,30,23)(20,33,34,40)(21,26,36,27)(24,39,41,35)(29,44,46,48)(31,42,32,38)(37,47,43,45);(1,3,7,4)(2,5,8,6)(9,10)(11,15)(12,14)(13,16)(17,19,22,30)(18,23,25,28)(20,29,34,46)(21,32,36,31)(24,37,41,43)(26,42,27,38)(33,48,40,44)(35,47,39,45);(1,4,7,3)(2,5,8,6)(9,11)(10,13)(12,16)(14,15)(17,20,21,35,22,34,36,39)(18,24,27,33,25,41,26,40)(19,29,32,47,30,46,31,45)(23,37,38,48,28,43,42,44);(1,5,7,6)(2,3,8,4)(11,16)(13,15)(18,25)(19,30)(20,34)(26,27)(31,32)(35,39)(37,43)(44,48);(1,6,7,5)(2,3,8,4)(10,14)(13,15)(17,21,22,36)(18,26,25,27)(19,31,30,32)(20,35,34,39)(23,38,28,42)(24,40,41,33)(29,45,46,47)(37,48,43,44)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fgi,d^2=fi,e^2=f,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, 6822867);