Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1688 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,6)(4,8)(5,7)(9,10,12,14)(11,15,16,13)(19,23)(22,24)(25,26)(27,33)(28,32)(29,30)(31,34)(35,40)(36,38)(37,39);(1,3,5,8)(2,6,7,4)(9,10,12,14)(11,13,16,15)(18,21)(22,24)(26,30)(27,34)(32,36)(35,37);(1,3)(2,4)(5,8)(6,7)(9,11)(10,13)(12,16)(14,15)(17,18,20,21)(19,22,23,24)(25,26)(27,31)(28,36)(29,30)(32,38)(33,34)(35,40)(37,39);(1,3)(2,4)(5,8)(6,7)(9,10)(11,13)(12,14)(15,16)(17,19,20,23)(18,22,21,24)(25,27,29,34)(26,31,30,33)(28,37,38,35)(32,40,36,39);(1,4)(2,3)(5,6)(7,8)(9,11)(10,13)(12,16)(14,15)(25,28)(26,32)(27,35)(29,38)(30,36)(31,39)(33,40)(34,37)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=fgi,d^2=gi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3454307);