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 | |
1665 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)(3,5)(4,6)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18,22,20)(19,24,21,23)(25,26,29,30)(27,33,34,31)(28,36,37,32)(35,38,40,39);(4,8)(6,7)(9,11)(10,15)(12,13)(14,16)(17,18,22,20)(19,23,21,24)(26,30)(27,34)(28,37)(38,39);(2,3)(6,7)(10,12)(11,16)(17,19,22,21)(18,23,20,24)(25,26)(27,31)(28,36)(29,30)(32,37)(33,34)(35,39)(38,40);(1,3,5,2)(4,7,8,6)(9,12,14,10)(11,13,16,15)(17,20,22,18)(19,23,21,24)(25,27,29,34)(26,31,30,33)(28,35,37,40)(32,38,36,39);(1,4)(2,6)(3,7)(5,8)(9,13)(10,11)(12,16)(14,15)(17,21,22,19)(18,23,20,24)(25,28)(26,32)(27,35)(29,37)(30,36)(31,38)(33,39)(34,40)Polycyclic presentation:
2^9|a^2=fgi,b^2=gi,c^2=gi,d^2=g,e^2=gi,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, 5108588);