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 | |
4566 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(10,14)(15,20)(16,21)(22,24)(25,26)(27,37)(28,40)(29,35)(30,34)(31,36)(32,39)(33,38);(1,3)(2,4)(5,8)(6,7)(9,10,13,14)(11,15,18,20)(12,16,19,21)(17,22,23,24)(25,27)(26,32)(28,29)(30,38)(31,39)(33,35)(34,40)(36,37);(1,3,5,8)(2,6,7,4)(12,19)(16,21)(17,23)(22,24)(25,28,31,38)(26,33,36,40)(27,29,39,30)(32,35,37,34);(9,10)(11,15)(12,16)(13,14)(17,22)(18,20)(19,21)(23,24)(25,29,31,30)(26,34,36,35)(27,28,39,38)(32,40,37,33);(1,4)(2,3)(5,6)(7,8)(9,11,13,18)(10,15,14,20)(12,17,19,23)(16,22,21,24)(25,30,31,29)(26,35,36,34)(27,38,39,28)(32,33,37,40);(1,3)(2,4)(5,8)(6,7)(9,12)(10,16)(11,17)(13,19)(14,21)(15,22)(18,23)(20,24)(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=ghi,c^2=h,d^2=hi,e^2=g,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, 8096603);