Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2735949 |
| 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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1574 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,8)(4,6)(5,7)(10,14)(11,15)(25,26,29,32)(27,33,34,30)(28,36,37,31)(35,38,40,39);(1,3)(2,6)(4,5)(7,8)(9,10,13,14)(11,16,15,12)(18,21)(22,24)(27,34)(30,33)(35,40)(38,39);(3,4)(6,8)(10,14)(11,15)(17,18,20,21)(19,22,23,24)(25,26)(27,30)(28,31)(29,32)(33,34)(35,38)(36,37)(39,40);(1,4,5,3)(2,6,7,8)(9,11,13,15)(10,12,14,16)(25,27)(26,30)(28,35)(29,34)(31,38)(32,33)(36,39)(37,40);(1,2)(3,6)(4,8)(5,7)(9,12)(10,11)(13,16)(14,15)(17,19,20,23)(18,22,21,24)(25,28,29,37)(26,31,32,36)(27,35,34,40)(30,38,33,39)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=fh,d^2=f,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,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, 2648439);