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 | |
4526 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,7)(9,10,15,17)(11,16,21,24)(12,18,19,13)(14,20,23,22)(25,26,31,34)(27,29,32,37)(28,30,33,39)(35,36,38,40);(1,2,5,4)(3,7,8,6)(9,11,15,21)(10,16,17,24)(12,20,19,22)(13,14,18,23)(25,27,31,32)(26,29,34,37)(28,35,33,38)(30,36,39,40);(1,3,5,8)(2,6,4,7)(10,17)(11,21)(13,18)(20,22)(27,32)(29,37)(35,38)(36,40);(1,4,5,2)(3,7,8,6)(9,12,15,19)(10,18,17,13)(11,22,21,20)(14,24,23,16)(25,28)(26,30)(27,35)(29,36)(31,33)(32,38)(34,39)(37,40);(9,13,15,18)(10,19,17,12)(11,23,21,14)(16,20,24,22)(25,29,31,37)(26,32,34,27)(28,36,33,40)(30,38,39,35);(9,14,15,23)(10,20,17,22)(11,18,21,13)(12,16,19,24)(25,30)(26,33)(27,36)(28,34)(29,38)(31,39)(32,40)(35,37)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=g,d^2=gh,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8529448);