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 | |
4448 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10,14,16)(11,15,19,22)(12,13,20,21)(17,18,23,24)(25,26)(27,30)(28,33)(29,34)(31,32)(35,40)(36,38)(37,39);(1,3,7,6)(2,4,8,5)(9,11,14,19)(10,15,16,22)(12,17,20,23)(13,18,21,24)(25,27,31,35)(26,30,32,40)(28,36,37,29)(33,38,39,34);(1,4,7,5)(2,3,8,6)(10,16)(12,20)(15,22)(17,23)(26,32)(27,35)(28,37)(34,38);(2,8)(4,5)(9,12)(10,13)(11,17)(14,20)(15,18)(16,21)(19,23)(22,24)(25,28)(26,33)(27,36)(29,35)(30,38)(31,37)(32,39)(34,40);(1,5,7,4)(2,3,8,6)(9,11)(10,15)(12,17)(13,18)(14,19)(16,22)(20,23)(21,24)(25,29,31,36)(26,34,32,38)(27,37,35,28)(30,39,40,33);(1,6,7,3)(2,5,8,4)(9,13)(10,12)(11,18)(14,21)(15,17)(16,20)(19,24)(22,23)(25,30)(26,35)(27,32)(28,38)(29,33)(31,40)(34,37)(36,39)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=g,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10054202);