Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4928752 |
| 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 | |
1630 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)(9,10,14,15)(11,16,13,12)(17,18,22,20)(19,23,21,24)(25,26)(27,31)(28,35)(29,30)(32,36)(33,37)(34,40)(38,39);(9,11,14,13)(10,12,15,16)(18,20)(23,24)(26,30)(27,33)(32,35)(34,39);(10,15)(12,16)(17,19,22,21)(18,23,20,24)(25,27,29,33)(26,31,30,37)(28,34,36,39)(32,38,35,40);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,15)(14,16)(17,20,22,18)(19,24,21,23)(25,26)(27,31)(28,32)(29,30)(33,37)(34,38)(35,36)(39,40);(1,3,4,7)(2,6,5,8)(9,13,14,11)(10,12,15,16)(17,21,22,19)(18,24,20,23)(25,28,29,36)(26,32,30,35)(27,34,33,39)(31,38,37,40)Polycyclic presentation:
2^9|a^2=fh,b^2=g,c^2=fg,d^2=fgh,e^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,d^c=dg104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4839293);