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 | |
4446 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(3,4)(9,10,15,13)(11,18,16,12)(14,17,22,21)(19,24,23,20)(25,26,31,34)(27,30,33,40)(28,29,32,39)(35,36,37,38);(1,2,6,5)(3,8,4,7)(9,11,15,16)(10,12,13,18)(14,19,22,23)(17,20,21,24)(25,27)(26,30)(28,37)(29,38)(31,33)(32,35)(34,40)(36,39);(1,3,6,4)(2,7,5,8)(14,22)(17,21)(19,23)(20,24)(27,33)(30,40)(35,37)(36,38);(1,4,6,3)(2,8,5,7)(9,12,15,18)(10,16,13,11)(14,20,22,24)(17,23,21,19)(25,28)(26,29)(27,35)(30,36)(31,32)(33,37)(34,39)(38,40);(1,5,6,2)(3,8,4,7)(9,13,15,10)(11,12,16,18)(14,21,22,17)(19,20,23,24)(25,29)(26,32)(27,36)(28,34)(30,37)(31,39)(33,38)(35,40);(9,14,15,22)(10,17,13,21)(11,19,16,23)(12,20,18,24)(25,30,31,40)(26,33,34,27)(28,36,32,38)(29,37,39,35)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=h,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8306694);