Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
1438 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)(4,8)(9,10,14,12)(11,15,13,16)(17,18,20,22)(19,23,24,21)(25,26)(27,31)(28,30)(29,36)(32,37)(33,39)(34,35)(38,40);(1,2,5,6)(3,8,7,4)(9,11,14,13)(10,15,12,16)(26,28)(31,35)(32,36)(38,39);(1,3)(2,4)(5,7)(6,8)(9,12,14,10)(11,15,13,16)(25,27,30,34)(26,31,28,35)(29,33,37,40)(32,38,36,39);(1,2)(3,4)(5,6)(7,8)(9,13,14,11)(10,15,12,16)(17,19,20,24)(18,21,22,23)(25,28,30,26)(27,31,34,35)(29,32,37,36)(33,39,40,38);(1,4)(2,3)(5,8)(6,7)(25,29,30,37)(26,32,28,36)(27,33,34,40)(31,38,35,39)Polycyclic presentation:
2^9|a^2=fi,b^2=h,c^2=fgh,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, 2685670);