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 | |
4517 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)(9,10,14,18)(11,15,19,22)(12,16,21,24)(13,17,23,20)(25,26)(27,36)(28,30)(29,33)(31,35)(32,37)(34,40)(38,39);(2,6)(3,7)(9,11)(10,15)(12,20)(13,16)(14,19)(17,21)(18,22)(23,24)(27,37)(29,39)(32,36)(33,38);(1,3,5,7)(2,4,6,8)(11,19)(12,21)(15,22)(16,24)(25,27,31,37)(26,32,35,36)(28,39,34,29)(30,33,40,38);(1,4)(2,7)(3,6)(5,8)(9,12)(10,16)(11,20)(13,15)(14,21)(17,19)(18,24)(22,23)(25,28,31,34)(26,30,35,40)(27,29,37,39)(32,38,36,33);(9,13)(10,17)(11,16)(12,22)(14,23)(15,21)(18,20)(19,24)(25,29)(26,33)(27,28)(30,36)(31,39)(32,40)(34,37)(35,38);(9,11)(10,15)(12,20)(13,16)(14,19)(17,21)(18,22)(23,24)(25,30)(26,34)(27,38)(28,35)(29,36)(31,40)(32,39)(33,37)Polycyclic presentation:
2^9|a^2=ghi,c^2=gh,d^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8812388);