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 | |
4486 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)(25,26)(27,34)(28,33)(29,30)(31,32)(35,40)(36,37)(38,39);(1,2,5,4)(3,7,8,6)(9,10,14,15)(11,13,17,22)(12,16,21,23)(18,20,19,24)(26,32)(29,35)(33,34)(36,38);(1,3,5,8)(2,6,4,7)(10,15)(13,22)(16,23)(20,24)(25,27,31,28)(26,33,32,34)(29,36,35,38)(30,39,40,37);(1,4,5,2)(3,7,8,6)(9,11,14,17)(10,13,15,22)(12,19,21,18)(16,24,23,20)(25,28,31,27)(26,34,32,33)(29,36,35,38)(30,39,40,37);(9,12,14,21)(10,16,15,23)(11,18,17,19)(13,20,22,24)(25,29,31,35)(26,30,32,40)(27,36,28,38)(33,39,34,37);(9,13)(10,17)(11,15)(12,20)(14,22)(16,19)(18,23)(21,24)(25,30)(26,35)(27,37)(28,39)(29,32)(31,40)(33,36)(34,38)Polycyclic presentation:
2^9|b^2=i,c^2=g,d^2=hi,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8809698);