Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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 | |
1438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,4)(6,7)(9,10)(11,16)(12,15)(13,14)(17,18,22,21)(19,23,20,24)(28,35)(32,37)(33,39)(38,40);(1,2,5,4)(3,7,8,6)(9,11)(10,12)(13,16)(14,15)(17,19,22,20)(18,23,21,24)(26,30)(31,36)(32,37)(38,40);(1,3)(2,6)(4,7)(5,8)(9,12)(10,15)(11,13)(14,16)(17,20,22,19)(18,23,21,24)(25,26)(27,31)(28,32)(29,30)(33,38)(34,36)(35,37)(39,40);(1,3)(2,7)(4,6)(5,8)(9,12)(10,11)(13,15)(14,16)(17,21,22,18)(19,23,20,24)(25,27,29,34)(26,31,30,36)(28,33,35,39)(32,38,37,40);(1,4,5,2)(3,7,8,6)(9,13,14,10)(11,16,15,12)(25,28)(26,32)(27,33)(29,35)(30,37)(31,38)(34,39)(36,40)Polycyclic presentation:
2^9|a^2=g,b^2=fgi,c^2=g,d^2=fgh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4778910);