Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #46366 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4234 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,7,5)(3,4,8,6)(11,17)(15,18)(19,23)(22,24)(25,26,31,28)(27,30,35,32)(29,33,38,36)(34,37,40,39);(2,5)(3,8)(9,10)(11,18)(12,20)(13,14)(15,17)(16,21)(19,22)(23,24)(25,26,31,28)(27,32,35,30)(29,36,38,33)(34,37,40,39);(1,3)(2,6)(4,5)(7,8)(10,14)(15,18)(16,20)(22,24)(25,27,31,35)(26,32,28,30)(29,34,38,40)(33,39,36,37);(1,4,7,6)(2,3,5,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,28,31,26)(27,30,35,32)(29,36,38,33)(34,37,40,39);(1,5,7,2)(3,4,8,6)(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,24)(21,23)(25,29)(26,33)(27,34)(28,36)(30,37)(31,38)(32,39)(35,40);(1,6,7,4)(2,8,5,3)(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,30)(26,27)(28,35)(29,37)(31,32)(33,34)(36,40)(38,39)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=gh,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8659483);