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 | |
4433 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(11,19)(12,21)(15,22)(16,24)(25,26)(27,37)(28,40)(29,35)(30,34)(31,36)(32,39)(33,38);(1,2,5,4)(3,6,8,7)(9,10,14,18)(11,15,19,22)(12,16,21,24)(13,17,20,23)(25,27)(26,32)(28,29)(30,38)(31,39)(33,35)(34,40)(36,37);(2,4)(6,7)(9,11)(10,15)(12,20)(13,21)(14,19)(16,23)(17,24)(18,22)(25,28,31,38)(26,33,36,40)(27,29,39,30)(32,35,37,34);(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,29,31,30)(26,34,36,35)(27,28,39,38)(32,40,37,33);(1,3,5,8)(2,6,4,7)(9,12,14,21)(10,16,18,24)(11,20,19,13)(15,23,22,17)(25,30,31,29)(26,35,36,34)(27,38,39,28)(32,33,37,40);(1,4,5,2)(3,7,8,6)(9,13,14,20)(10,17,18,23)(11,21,19,12)(15,24,22,16)(25,26)(27,32)(28,40)(29,34)(30,35)(31,36)(33,38)(37,39)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=g,d^2=g,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8087642);