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 | |
4422 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(4,8)(6,7)(9,10)(11,12)(13,21)(14,15)(16,17)(18,22)(19,23)(20,24)(26,32)(28,30)(33,40)(36,38);(1,2,5,3)(4,6,8,7)(10,15)(11,17)(18,21)(19,24)(25,26)(27,28)(29,33)(30,34)(31,32)(35,36)(37,38)(39,40);(2,3)(6,7)(10,15)(12,16)(18,21)(20,23)(25,27,31,34)(26,28,32,30)(29,37,39,35)(33,38,40,36);(9,11,14,17)(10,16,15,12)(13,19,22,24)(18,23,21,20)(25,28,31,30)(26,27,32,34)(29,36,39,38)(33,35,40,37);(1,3,5,2)(4,7,8,6)(9,12,14,16)(10,17,15,11)(13,20,22,23)(18,24,21,19)(25,29,31,39)(26,33,32,40)(27,35,34,37)(28,36,30,38);(1,4)(2,6)(3,7)(5,8)(9,13,14,22)(10,18,15,21)(11,19,17,24)(12,20,16,23)(25,30,31,28)(26,34,32,27)(29,38,39,36)(33,37,40,35)Polycyclic presentation:
2^9|b^2=i,c^2=gh,d^2=g,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8027206);