Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1648 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(10,14)(11,15)(17,18)(19,21)(20,22)(23,24)(25,26,30,31)(27,29,34,32)(28,33,37,39)(35,38,40,36);(2,3)(6,7)(9,10)(11,16)(12,15)(13,14)(17,18,20,22)(19,21,24,23)(26,31)(27,34)(33,39)(35,40);(4,8)(6,7)(10,14)(12,16)(17,18)(19,23)(20,22)(21,24)(25,27,30,34)(26,32,31,29)(28,35,37,40)(33,36,39,38);(1,3,5,2)(4,6,8,7)(9,11)(10,12)(13,15)(14,16)(17,18)(19,23)(20,22)(21,24)(25,28)(26,33)(27,35)(29,38)(30,37)(31,39)(32,36)(34,40);(1,4)(2,6)(3,7)(5,8)(9,12)(10,15)(11,14)(13,16)(17,19)(18,21)(20,24)(22,23)(25,29,30,32)(26,27,31,34)(28,36,37,38)(33,40,39,35)Polycyclic presentation:
2^9|a^2=g,b^2=fgi,c^2=g,d^2=fh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3281619);