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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1578 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,6)(3,4,7,8)(10,13)(14,16)(17,18)(19,21)(20,22)(23,24)(25,26)(27,31)(28,30)(29,36)(32,38)(33,35)(34,40)(37,39);(1,2,5,6)(3,4,7,8)(11,15)(14,16)(17,19,20,23)(18,21,22,24)(26,28)(27,33)(32,36)(34,37);(2,6)(3,7)(9,10,12,13)(11,14,15,16)(17,18,20,22)(19,21,23,24)(25,27,30,33)(26,31,28,35)(29,37,38,34)(32,40,36,39);(1,3)(2,4)(5,7)(6,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,21)(20,22)(23,24)(25,28,30,26)(27,31,33,35)(29,36,38,32)(34,39,37,40);(1,4)(2,3)(5,8)(6,7)(9,11)(10,14)(12,15)(13,16)(17,19)(18,21)(20,23)(22,24)(25,29)(26,32)(27,34)(28,36)(30,38)(31,39)(33,37)(35,40)Polycyclic presentation:
2^9|a^2=fh,b^2=fgi,c^2=gh,d^2=f,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, 4957526);