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 |
| prime radical | #35392 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4545 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)(3,4)(5,6)(7,8)(9,10,13,16)(11,14,17,21)(12,15,20,19)(18,22,23,24)(25,26)(27,28)(29,36)(30,33)(31,32)(34,35)(37,38)(39,40);(2,6)(3,7)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,20)(23,24)(25,27,31,29)(26,28,32,36)(30,34,39,37)(33,35,40,38);(1,3,5,7)(2,4,6,8)(11,17)(14,21)(18,23)(22,24)(26,32)(27,29)(33,40)(34,37);(1,4)(2,7)(3,6)(5,8)(9,10)(11,14)(12,19)(13,16)(15,20)(17,21)(18,24)(22,23)(25,28)(26,27)(29,32)(30,38)(31,36)(33,37)(34,40)(35,39);(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,29,31,27)(26,28,32,36)(30,37,39,34)(33,35,40,38);(9,12)(10,15)(11,18)(13,20)(14,22)(16,19)(17,23)(21,24)(25,30)(26,33)(27,34)(28,35)(29,37)(31,39)(32,40)(36,38)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=hi,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8436150);