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 | |
4520 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10,14,16)(11,15,20,19)(12,17,23,22)(13,18,24,21)(25,26)(27,34)(28,37)(29,40)(30,36)(31,35)(32,38)(33,39);(1,3,5,8)(2,6,7,4)(9,11)(10,15)(12,21)(13,17)(14,20)(16,19)(18,23)(22,24)(25,27,31,38)(26,32,35,34)(28,36,39,29)(30,37,40,33);(1,2)(3,4)(5,7)(6,8)(10,16)(11,20)(17,22)(18,21)(25,28,31,39)(26,33,35,37)(27,36,38,29)(30,34,40,32);(1,4)(2,3)(5,6)(7,8)(9,12)(10,17)(11,21)(13,15)(14,23)(16,22)(18,20)(19,24)(25,29)(26,30)(27,28)(31,36)(32,37)(33,34)(35,40)(38,39);(1,3)(2,4)(5,8)(6,7)(9,13,14,24)(10,18,16,21)(11,22,20,17)(12,19,23,15)(25,27)(26,34)(28,36)(29,39)(30,37)(31,38)(32,35)(33,40);(1,4)(2,3)(5,6)(7,8)(9,11)(10,19)(12,21)(13,22)(14,20)(15,16)(17,24)(18,23)(25,30)(26,29)(27,37)(28,32)(31,40)(33,38)(34,39)(35,36)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=hi,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8805859);