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 | |
4605 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)(9,10,14,17)(11,12,19,15)(13,16,21,23)(18,20,24,22)(25,26,31,33)(27,35,38,32)(28,39,37,30)(29,40,36,34);(2,6)(3,7)(9,11,14,19)(10,15,17,12)(13,18,21,24)(16,22,23,20)(25,27,31,38)(26,32,33,35)(28,36,37,29)(30,34,39,40);(1,3)(2,4)(5,7)(6,8)(13,21)(16,23)(18,24)(20,22)(26,33)(30,39)(32,35)(34,40);(1,4)(2,3)(5,8)(6,7)(9,12,14,15)(10,11,17,19)(13,20,21,22)(16,18,23,24)(25,28)(26,30)(27,36)(29,38)(31,37)(32,34)(33,39)(35,40);(9,11)(10,12)(13,18)(14,19)(15,17)(16,20)(21,24)(22,23)(25,29,31,36)(26,34,33,40)(27,37,38,28)(30,35,39,32);(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,30)(26,28)(27,34)(29,35)(31,39)(32,36)(33,37)(38,40)Polycyclic presentation:
2^9|a^2=g,b^2=gi,d^2=ghi,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8300828);