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 | |
4578 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,7,5,8)(9,10)(11,18)(12,13)(14,15)(16,19)(17,22)(20,21)(23,24)(25,26,31,28)(27,32,34,30)(29,33,39,36)(35,40,37,38);(1,3,6,5)(2,7,4,8)(10,15)(11,19)(12,22)(21,24)(25,27,31,34)(26,32,28,30)(29,37,39,35)(33,38,36,40);(1,3,6,5)(2,8,4,7)(10,15)(13,17)(16,18)(21,24)(27,34)(30,32)(35,37)(38,40);(1,4,6,2)(3,7,5,8)(9,11,14,19)(10,16,15,18)(12,23,22,20)(13,21,17,24)(25,28,31,26)(27,30,34,32)(29,33,39,36)(35,40,37,38);(1,5,6,3)(2,7,4,8)(9,12)(10,17)(11,20)(13,15)(14,22)(16,21)(18,24)(19,23)(25,29,31,39)(26,33,28,36)(27,35,34,37)(30,38,32,40);(9,13,14,17)(10,12,15,22)(11,21,19,24)(16,23,18,20)(25,30)(26,27)(28,34)(29,38)(31,32)(33,35)(36,37)(39,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,c^2=gh,d^2=hi,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,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, 8278488);