Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1412 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(6,8)(10,13)(11,12)(19,23)(22,24)(25,26,30,34)(27,31,32,28)(29,33,38,40)(35,39,37,36);(9,10)(11,15)(12,16)(13,14)(18,21)(22,24)(25,27,30,32)(26,28,34,31)(29,35,38,37)(33,36,40,39);(9,11,14,12)(10,15,13,16)(17,18,20,21)(19,22,23,24)(25,27)(26,31)(28,34)(29,37)(30,32)(33,36)(35,38)(39,40);(1,2)(3,6)(4,5)(7,8)(9,12,14,11)(10,15,13,16)(25,28,30,31)(26,32,34,27)(29,36,38,39)(33,37,40,35);(1,3,4,7)(2,6,5,8)(9,13,14,10)(11,15,12,16)(17,19,20,23)(18,22,21,24)(25,29)(26,33)(27,35)(28,36)(30,38)(31,39)(32,37)(34,40)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,c^2=fh,d^2=f,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=efh,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, 5270474);