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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1896 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,6)(3,7,8,4)(9,10)(11,12)(13,15)(14,16)(18,22)(20,24)(25,26,30,31)(27,29,34,32)(28,33,37,39)(35,38,40,36);(1,3,5,8)(2,4,6,7)(9,11,14,15)(10,13,16,12)(17,18,21,22)(19,20,23,24)(26,31)(27,34)(33,39)(35,40);(3,8)(4,7)(9,12,14,13)(10,11,16,15)(17,19,21,23)(18,20,22,24)(25,27,30,34)(26,32,31,29)(28,35,37,40)(33,36,39,38);(1,2)(3,7)(4,8)(5,6)(9,13,14,12)(10,15,16,11)(17,20,21,24)(18,19,22,23)(25,28)(26,33)(27,35)(29,38)(30,37)(31,39)(32,36)(34,40);(1,4,5,7)(2,3,6,8)(9,10)(11,13)(12,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,29,30,32)(26,27,31,34)(28,36,37,38)(33,40,39,35)Polycyclic presentation:
2^9|a^2=fghi,b^2=fg,c^2=hi,d^2=ghi,e^2=fghi,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, 3763036);