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 | |
1614 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)(3,8,4,7)(9,10)(11,15)(12,13)(14,16)(19,23)(22,24)(26,30)(27,33)(32,37)(35,39);(2,6)(3,4)(9,11,13,16)(10,14,12,15)(18,21)(22,24)(28,36)(32,37)(35,39)(38,40);(1,3,5,4)(2,7,6,8)(9,10,13,12)(11,14,16,15)(17,18)(19,22)(20,21)(23,24)(25,26)(27,34)(28,32)(29,30)(31,33)(35,40)(36,37)(38,39);(1,4,5,3)(2,7,6,8)(9,12,13,10)(11,14,16,15)(25,27)(26,31)(28,35)(29,33)(30,34)(32,38)(36,39)(37,40);(1,2)(3,8)(4,7)(5,6)(9,11)(10,15)(12,14)(13,16)(17,19)(18,22)(20,23)(21,24)(25,28,29,36)(26,32,30,37)(27,35,33,39)(31,38,34,40)Polycyclic presentation:
2^9|a^2=gi,b^2=fgi,c^2=f,d^2=f,e^2=fg,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, 4971033);