Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6132643 |
| 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 | |
1368 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,15)(12,16)(13,14)(19,23)(22,24)(25,26,31,28)(27,33,29,32)(30,34,38,36)(35,40,37,39);(3,8)(5,7)(10,14)(12,15)(18,21)(19,23)(25,27,31,29)(26,32,28,33)(30,35,38,37)(34,39,36,40);(10,14)(12,15)(17,18,20,21)(19,22,23,24)(25,28,31,26)(27,32,29,33)(30,34,38,36)(35,40,37,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,12)(13,16)(14,15)(17,19,20,23)(18,22,21,24)(25,29,31,27)(26,33,28,32)(30,37,38,35)(34,40,36,39);(1,2)(3,5)(4,6)(7,8)(9,12)(10,11)(13,15)(14,16)(25,30,31,38)(26,34,28,36)(27,35,29,37)(32,39,33,40)Polycyclic presentation:
2^9|a^2=fhi,b^2=fg,c^2=h,d^2=h,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dfi,e^b=eg,d^c=dfg,e^c=ef62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6082192);