Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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 | |
1376 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,6,7)(3,8,4,5)(9,10,14,16)(11,12,15,13)(19,23)(21,24)(25,26,29,32)(27,33,34,30)(28,31,36,38)(35,39,40,37);(2,7)(3,4)(9,11)(10,13)(12,16)(14,15)(17,18,20,22)(19,21,23,24)(27,34)(28,36)(30,33)(31,38);(1,3,6,4)(2,5,7,8)(9,12,14,13)(10,15,16,11)(25,27)(26,30)(28,35)(29,34)(31,37)(32,33)(36,40)(38,39);(1,4,6,3)(2,5,7,8)(9,11)(10,12)(13,16)(14,15)(17,19)(18,21)(20,23)(22,24)(25,28,29,36)(26,31,32,38)(27,35,34,40)(30,37,33,39);(1,5)(2,4)(3,7)(6,8)(9,13,14,12)(10,15,16,11)(25,26)(27,30)(28,31)(29,32)(33,34)(35,37)(36,38)(39,40)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=f,d^2=g,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4781352);