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 | |
4773 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10,14,15)(11,16,19,23)(12,20,21,13)(17,18,22,24)(25,26,30,34)(27,28,32,31)(29,33,37,40)(35,36,39,38);(2,6)(3,7)(10,15)(12,21)(16,23)(17,22)(25,27,30,32)(26,31,34,28)(29,35,37,39)(33,38,40,36);(1,3)(2,4)(5,7)(6,8)(10,15)(13,20)(16,23)(18,24)(29,37)(33,40)(35,39)(36,38);(1,4)(2,3)(5,8)(6,7)(9,11,14,19)(10,16,15,23)(12,22,21,17)(13,18,20,24)(25,27)(26,28)(29,35)(30,32)(31,34)(33,36)(37,39)(38,40);(9,12)(10,13)(11,17)(14,21)(15,20)(16,24)(18,23)(19,22)(25,28)(26,32)(27,34)(29,36)(30,31)(33,39)(35,40)(37,38);(9,13)(10,12)(11,18)(14,20)(15,21)(16,22)(17,23)(19,24)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,d^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8313989);