Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | C22×#121 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1840 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)(3,4)(5,6)(7,8)(9,10)(11,16)(12,15)(13,14)(17,18,20,22)(19,21,23,24)(25,26,29,32)(27,33,34,30)(28,36,37,31)(35,38,40,39);(2,6)(4,7)(10,14)(12,16)(17,19,20,23)(18,21,22,24)(27,34)(28,37)(30,33)(31,36);(1,3)(2,7)(4,6)(5,8)(9,10,13,14)(11,12,15,16)(17,19)(18,21)(20,23)(22,24)(25,27)(26,30)(28,35)(29,34)(31,38)(32,33)(36,39)(37,40);(1,3)(2,4)(5,8)(6,7)(9,11)(10,12)(13,15)(14,16)(25,26)(27,33)(28,31)(29,32)(30,34)(35,39)(36,37)(38,40);(1,4)(2,8)(3,6)(5,7)(9,12)(10,15)(11,14)(13,16)(17,18)(19,21)(20,22)(23,24)(25,28)(26,31)(27,35)(29,37)(30,38)(32,36)(33,39)(34,40)Polycyclic presentation:
2^9|a^2=fghi,b^2=hi,c^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5690952);