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 | |
4612 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,14)(12,19)(13,15)(16,21)(17,23)(18,24)(20,22)(25,26,31,30)(27,34,29,32)(28,36,38,33)(35,39,37,40);(1,2,5,7)(3,4,8,6)(9,11,13,17)(10,14,15,23)(12,20,21,18)(16,24,19,22)(25,27,31,29)(26,32,30,34)(28,35,38,37)(33,39,36,40);(1,3,5,8)(2,6,7,4)(10,15)(11,17)(16,19)(18,20)(25,26,31,30)(27,32,29,34)(28,33,38,36)(35,39,37,40);(9,10)(11,14)(12,16)(13,15)(17,23)(18,22)(19,21)(20,24)(25,28)(26,33)(27,35)(29,37)(30,36)(31,38)(32,39)(34,40);(1,3)(2,4)(5,8)(6,7)(9,11)(10,14)(12,18)(13,17)(15,23)(16,22)(19,24)(20,21)(25,29,31,27)(26,32,30,34)(28,37,38,35)(33,39,36,40);(1,4)(2,3)(5,6)(7,8)(9,12)(10,16)(11,18)(13,21)(14,22)(15,19)(17,20)(23,24)(25,30,31,26)(27,32,29,34)(28,36,38,33)(35,39,37,40)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=ghi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9156689);