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 | |
4323 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)(9,10,13,15)(11,16,19,14)(12,20,22,17)(18,23,21,24)(25,26,31,29)(27,35,36,32)(28,37,38,33)(30,34,40,39);(2,6)(3,7)(9,11,13,19)(10,14,15,16)(12,18,22,21)(17,23,20,24)(25,27)(26,32)(28,30)(29,35)(31,36)(33,34)(37,39)(38,40);(1,3,5,7)(2,4,6,8)(10,15)(14,16)(17,20)(23,24)(25,26,31,29)(27,32,36,35)(28,33,38,37)(30,34,40,39);(1,4)(2,7)(3,6)(5,8)(9,10)(11,14)(12,17)(13,15)(16,19)(18,23)(20,22)(21,24)(25,28)(26,33)(27,30)(29,37)(31,38)(32,34)(35,39)(36,40);(9,11)(10,16)(12,21)(13,19)(14,15)(17,23)(18,22)(20,24)(25,29,31,26)(27,35,36,32)(28,33,38,37)(30,34,40,39);(9,12)(10,17)(11,18)(13,22)(14,23)(15,20)(16,24)(19,21)(25,30)(26,34)(27,28)(29,39)(31,40)(32,33)(35,37)(36,38)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=gh,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10055059);