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 | |
1616 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)(9,10,13,14)(11,16,12,15)(17,18,21,19)(20,23,24,22)(25,26)(27,28)(29,33)(30,31)(32,35)(34,36)(37,40)(38,39);(1,3,5,8)(2,4,7,6)(10,14)(11,12)(17,19,21,18)(20,23,24,22)(26,31)(28,35)(33,39)(36,40);(3,8)(4,6)(9,11,13,12)(10,15,14,16)(25,27)(26,28)(29,37)(30,32)(31,35)(33,40)(34,38)(36,39);(1,3)(2,6)(4,7)(5,8)(9,12,13,11)(10,15,14,16)(17,20)(18,22)(19,23)(21,24)(25,28)(26,32)(27,31)(29,36)(30,35)(33,37)(34,39)(38,40);(1,4,5,6)(2,3,7,8)(9,10)(11,16)(12,15)(13,14)(25,29)(26,33)(27,34)(28,36)(30,38)(31,39)(32,37)(35,40)Polycyclic presentation:
2^9|a^2=fghi,b^2=fgi,c^2=g,d^2=g,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, 4980974);