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 | |
1734 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,6,8)(3,5,4,7)(10,12)(11,15)(17,18,20,21)(19,23,24,22)(25,26)(27,32)(28,33)(29,34)(30,31)(35,40)(36,37)(38,39);(1,3)(2,7)(4,6)(5,8)(9,10,13,12)(11,16,15,14)(18,21)(19,24)(26,31)(28,32)(34,39)(36,40);(3,4)(5,7)(10,12)(11,15)(17,19,20,24)(18,22,21,23)(25,27)(26,32)(28,31)(29,37)(30,33)(34,36)(35,38)(39,40);(1,4,6,3)(2,7,8,5)(9,11,13,15)(10,14,12,16)(17,18)(19,22)(20,21)(23,24)(25,28,30,32)(26,33,31,27)(29,36,38,40)(34,37,39,35);(1,5,6,7)(2,3,8,4)(9,12,13,10)(11,16,15,14)(17,19)(18,22)(20,24)(21,23)(25,29)(26,34)(27,35)(28,36)(30,38)(31,39)(32,40)(33,37)Polycyclic presentation:
2^9|a^2=ghi,b^2=gi,c^2=g,d^2=fg,e^2=gh,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, 4980979);