Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4462490 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1398 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,7)(3,4,8,6)(10,13)(14,16)(19,23)(21,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,36)(37,39)(38,40);(1,2)(3,4)(5,7)(6,8)(11,15)(14,16)(17,18,20,22)(19,21,23,24)(26,31)(28,34)(32,39)(35,38);(1,3,5,8)(2,6,7,4)(9,10)(11,14)(12,13)(15,16)(25,27,30,37)(26,32,31,39)(28,38,34,35)(29,36,33,40);(1,3)(2,4)(5,8)(6,7)(9,11,12,15)(10,14,13,16)(17,19)(18,21)(20,23)(22,24)(25,28,30,34)(26,33,31,29)(27,35,37,38)(32,36,39,40);(1,4)(2,3)(5,6)(7,8)(25,29,30,33)(26,34,31,28)(27,36,37,40)(32,35,39,38)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=f,d^2=fg,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfg,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4371186);