Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| 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 | |
1478 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(5,8)(11,15)(14,16)(17,18,20,22)(19,23,24,21)(25,26)(27,31)(28,36)(29,30)(32,38)(33,34)(35,40)(37,39);(1,2,4,6)(3,5,7,8)(10,13)(14,16)(26,30)(27,33)(32,36)(35,37);(1,3)(2,5)(4,7)(6,8)(9,10,12,13)(11,14,15,16)(25,26)(27,34)(28,32)(29,30)(31,33)(35,40)(36,38)(37,39);(1,2)(3,5)(4,6)(7,8)(9,11,12,15)(10,14,13,16)(17,19,20,24)(18,21,22,23)(25,27)(26,31)(28,37)(29,33)(30,34)(32,40)(35,38)(36,39);(1,3)(2,5)(4,7)(6,8)(25,28,29,38)(26,32,30,36)(27,35,33,37)(31,39,34,40)Polycyclic presentation:
2^9|a^2=fhi,b^2=g,c^2=fh,d^2=i,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2685522);