Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4379698 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1446 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)(3,7)(4,8)(5,6)(11,15)(13,16)(17,18,20,22)(19,23,24,21)(25,26)(27,33)(28,30)(29,32)(31,35)(34,40)(36,38)(37,39);(1,2,5,6)(3,7,8,4)(26,28)(31,33)(32,36)(39,40);(2,6)(4,7)(9,10,12,14)(11,13,15,16)(17,18)(19,21)(20,22)(23,24)(25,27)(26,31)(28,33)(29,37)(30,35)(32,40)(34,38)(36,39);(1,3)(2,4)(5,8)(6,7)(9,11,12,15)(10,13,14,16)(17,19,20,24)(18,21,22,23)(25,28,30,26)(27,33,35,31)(29,36,38,32)(34,40,37,39);(1,4)(2,3)(5,7)(6,8)(25,29,30,38)(26,32,28,36)(27,34,35,37)(31,39,33,40)Polycyclic presentation:
2^9|a^2=fg,b^2=h,c^2=gi,d^2=hi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4304889);