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 | |
1628 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(3,7)(11,15)(14,16)(19,23)(21,24)(25,26,29,33)(27,31,35,30)(28,32,38,37)(34,40,36,39);(3,7)(6,8)(10,13)(14,16)(25,27,29,35)(26,30,33,31)(28,34,38,36)(32,39,37,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18,20,22)(19,21,23,24)(25,27)(26,31)(28,36)(29,35)(30,33)(32,39)(34,38)(37,40);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,15)(13,16)(17,19,20,23)(18,21,22,24)(25,26)(27,31)(28,37)(29,33)(30,35)(32,38)(34,39)(36,40);(1,2)(3,6)(4,5)(7,8)(25,28)(26,32)(27,34)(29,38)(30,39)(31,40)(33,37)(35,36)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=ghi,d^2=hi,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, 3443436);