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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1693 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(6,8)(10,12)(11,15)(25,26)(27,31)(28,32)(29,30)(33,34)(35,39)(36,40)(37,38);(9,10,13,12)(11,14,15,16)(18,21)(22,24)(26,30)(27,33)(32,38)(35,36);(1,2,5,4)(3,6,7,8)(9,11)(10,14)(12,16)(13,15)(17,18,20,21)(19,22,23,24)(25,26)(27,34)(28,32)(29,30)(31,33)(35,40)(36,39)(37,38);(1,3)(2,6)(4,8)(5,7)(9,12,13,10)(11,14,15,16)(25,27,29,33)(26,31,30,34)(28,36,37,35)(32,40,38,39);(1,4,5,2)(3,8,7,6)(9,11)(10,14)(12,16)(13,15)(17,19,20,23)(18,22,21,24)(25,28,29,37)(26,32,30,38)(27,35,33,36)(31,39,34,40)Polycyclic presentation:
2^9|b^2=g,c^2=fhi,d^2=f,e^2=ghi,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, 2666349);