Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
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 | |
1554 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,6)(4,7)(9,10,13,15)(11,14,12,16)(18,21)(22,24)(25,26,31,29)(27,33,28,32)(30,34,38,37)(35,40,36,39);(1,2)(3,7)(4,8)(5,6)(9,10)(11,16)(12,14)(13,15)(25,27,31,28)(26,32,29,33)(30,35,38,36)(34,39,37,40);(1,3)(2,4)(5,8)(6,7)(9,11,13,12)(10,14,15,16)(17,18,20,21)(19,22,23,24)(25,28,31,27)(26,33,29,32)(30,36,38,35)(34,40,37,39);(1,3)(2,7)(4,6)(5,8)(9,10)(11,14)(12,16)(13,15)(17,19,20,23)(18,22,21,24)(25,29,31,26)(27,33,28,32)(30,34,38,37)(35,39,36,40);(1,4,5,7)(2,8,6,3)(9,12,13,11)(10,14,15,16)(25,30)(26,34)(27,35)(28,36)(29,37)(31,38)(32,39)(33,40)Polycyclic presentation:
2^9|a^2=fhi,b^2=fh,c^2=gi,d^2=g,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4340874);