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 | |
4632 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)(3,4)(5,6)(7,8)(10,11)(15,17)(16,18)(22,23)(25,26)(27,35)(28,37)(29,34)(30,32)(31,38)(33,39)(36,40);(2,6)(3,7)(9,10,14,11)(12,17,20,15)(13,16,21,18)(19,23,24,22)(25,27,30,38)(26,31,32,35)(28,34,39,36)(29,37,40,33);(1,3,5,7)(2,4,6,8)(13,21)(16,18)(19,24)(22,23)(26,32)(29,40)(31,35)(33,37);(1,4)(2,7)(3,6)(5,8)(9,11,14,10)(12,17,20,15)(13,18,21,16)(19,23,24,22)(25,28,30,39)(26,33,32,37)(27,36,38,34)(29,31,40,35);(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,27)(26,31)(28,36)(29,37)(30,38)(32,35)(33,40)(34,39);(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(25,29,30,40)(26,34,32,36)(27,37,38,33)(28,31,39,35)Polycyclic presentation:
2^9|b^2=gi,c^2=i,d^2=gi,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8177755);