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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1822 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10,13,15)(11,16,12,14)(17,18)(19,20)(21,24)(22,23)(25,26,31,33)(27,29,28,35)(30,32,38,40)(34,37,36,39);(1,2)(3,7)(4,6)(5,8)(9,11)(10,14)(12,13)(15,16)(18,23)(19,21)(25,27,31,28)(26,29,33,35)(30,36,38,34)(32,39,40,37);(1,2)(3,5)(4,6)(7,8)(9,10)(11,16)(12,14)(13,15)(17,19,22,21)(18,24,23,20)(25,28,31,27)(26,29,33,35)(30,36,38,34)(32,37,40,39);(1,2)(3,7)(4,6)(5,8)(9,10)(11,14)(12,16)(13,15)(17,20)(18,19)(21,23)(22,24)(25,29)(26,28)(27,33)(30,37)(31,35)(32,36)(34,40)(38,39);(1,3,4,8)(2,5,6,7)(9,12,13,11)(10,14,15,16)(17,21,22,19)(18,24,23,20)(25,30,31,38)(26,32,33,40)(27,34,28,36)(29,37,35,39)Polycyclic presentation:
2^9|a^2=fgi,b^2=fg,c^2=gi,e^2=g,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, 5622918);