Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4033837 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
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 | |
1434 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(3,7)(9,10,13,12)(11,14,15,16)(17,18,22,20)(19,23,21,24)(25,26,29,30)(27,33,34,31)(28,36,37,32)(35,39,38,40);(2,5)(3,7)(11,15)(14,16)(17,19,22,21)(18,23,20,24)(26,30)(27,34)(28,37)(39,40);(1,2)(3,6)(4,5)(7,8)(9,11,13,15)(10,14,12,16)(17,20,22,18)(19,23,21,24)(25,26)(27,31)(28,36)(29,30)(32,37)(33,34)(35,40)(38,39);(9,12,13,10)(11,16,15,14)(17,21,22,19)(18,23,20,24)(25,27,29,34)(26,31,30,33)(28,38,37,35)(32,40,36,39);(1,3,4,7)(2,6,5,8)(25,28,29,37)(26,32,30,36)(27,35,34,38)(31,39,33,40)Polycyclic presentation:
2^9|a^2=fgh,b^2=fi,c^2=gh,d^2=fgh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3964856);