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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1766 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)(3,6)(4,8)(5,7)(9,10)(11,14)(12,15)(13,16)(17,18,20,21)(19,23,24,22)(25,26,29,32)(27,33,34,30)(28,36,37,31)(35,38,40,39);(1,3,5,8)(2,4,7,6)(9,10,13,16)(11,12,15,14)(18,21)(22,23)(27,34)(28,37)(30,33)(31,36);(3,8)(4,6)(11,15)(12,14)(17,18,20,21)(19,23,24,22)(25,27,29,34)(26,30,32,33)(28,35,37,40)(31,38,36,39);(1,3)(2,6)(4,7)(5,8)(9,11,13,15)(10,14,16,12)(17,18)(19,22)(20,21)(23,24)(25,26)(27,33)(28,31)(29,32)(30,34)(35,39)(36,37)(38,40);(1,4,5,6)(2,3,7,8)(9,12,13,14)(10,15,16,11)(17,19)(18,22)(20,24)(21,23)(25,28,29,37)(26,31,32,36)(27,35,34,40)(30,38,33,39)Polycyclic presentation:
2^9|a^2=i,b^2=fgi,c^2=i,d^2=fgh,e^2=fi,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, 5110677);