Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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)(3,8)(4,5)(6,7)(9,10,15,16)(11,12,13,14)(17,18,21,19)(20,23,24,22)(25,26,29,30)(27,31,35,33)(28,32,37,38)(34,39,36,40);(1,3)(2,4)(5,6)(7,8)(9,11,15,13)(10,12,16,14)(17,19,21,18)(20,23,24,22)(26,30)(31,33)(32,38)(39,40);(2,7)(3,5)(9,12,15,14)(10,11,16,13)(25,27,29,35)(26,31,30,33)(28,36,37,34)(32,40,38,39);(1,4,6,8)(2,3,7,5)(9,13,15,11)(10,12,16,14)(17,20)(18,22)(19,23)(21,24)(25,26)(27,33)(28,32)(29,30)(31,35)(34,40)(36,39)(37,38);(1,5,6,3)(2,8,7,4)(9,14,15,12)(10,11,16,13)(25,28,29,37)(26,32,30,38)(27,34,35,36)(31,39,33,40)Polycyclic presentation:
2^9|a^2=hi,b^2=fi,c^2=fg,d^2=g,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4980897);