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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4503 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)(9,10,14,15)(11,16,18,12)(13,21,23,17)(19,20,24,22)(25,26)(27,33)(28,39)(29,32)(30,34)(31,36)(35,38)(37,40);(2,5)(3,8)(10,15)(12,16)(17,21)(20,22)(25,27,32,40)(26,33,29,37)(28,35,34,36)(30,31,39,38);(1,3)(2,6)(4,5)(7,8)(10,15)(11,18)(17,21)(19,24)(25,28,32,34)(26,30,29,39)(27,36,40,35)(31,33,38,37);(1,4,7,6)(2,3,5,8)(9,11,14,18)(10,16,15,12)(13,19,23,24)(17,22,21,20)(25,29,32,26)(27,37,40,33)(28,30,34,39)(31,36,38,35);(1,5,7,2)(3,4,8,6)(9,12,14,16)(10,11,15,18)(13,22,23,20)(17,24,21,19)(25,30)(26,34)(27,38)(28,29)(31,40)(32,39)(33,35)(36,37);(1,6,7,4)(2,8,5,3)(9,13,14,23)(10,17,15,21)(11,19,18,24)(12,20,16,22)(25,31,32,38)(26,35,29,36)(27,39,40,30)(28,33,34,37)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=gh,d^2=h,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9076503);