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 | |
4680 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,7)(9,10)(11,17)(12,20)(13,14)(15,18)(16,22)(19,23)(21,24)(25,26,31,33)(27,36,37,32)(28,38,35,30)(29,34,39,40);(1,2,5,4)(3,7,8,6)(10,14)(15,17)(16,20)(23,24)(25,27)(26,32)(28,29)(30,34)(31,37)(33,36)(35,39)(38,40);(1,3,5,8)(2,6,4,7)(11,18)(15,17)(19,21)(23,24)(26,33)(27,37)(28,35)(34,40);(1,4,5,2)(3,7,8,6)(9,11,13,18)(10,15,14,17)(12,21,22,19)(16,24,20,23)(25,28,31,35)(26,30,33,38)(27,29,37,39)(32,34,36,40);(9,12)(10,16)(11,19)(13,22)(14,20)(15,23)(17,24)(18,21)(25,29)(26,34)(27,35)(28,37)(30,36)(31,39)(32,38)(33,40);(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,22)(21,24)(25,30)(26,35)(27,34)(28,33)(29,36)(31,38)(32,39)(37,40)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=hi,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dghi,e^d=egi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9297745);