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 | |
4578 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,11)(15,17)(16,18)(22,23)(25,26)(27,34)(28,36)(29,31)(30,32)(33,38)(35,39)(37,40);(1,2)(3,5)(4,6)(7,8)(9,10,14,11)(12,17,19,15)(13,16,21,18)(20,23,24,22)(25,27,31,30)(26,32,29,34)(28,37,38,35)(33,40,36,39);(2,3)(6,7)(12,19)(15,17)(20,24)(22,23)(26,29)(32,34)(33,36)(39,40);(4,8)(6,7)(9,11,14,10)(12,17,19,15)(13,16,21,18)(20,22,24,23)(25,28,31,38)(26,33,29,36)(27,35,30,37)(32,39,34,40);(1,3,5,2)(4,7,8,6)(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,29,31,26)(27,32,30,34)(28,36,38,33)(35,39,37,40);(1,4)(2,6)(3,7)(5,8)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,30,31,27)(26,32,29,34)(28,35,38,37)(33,40,36,39)Polycyclic presentation:
2^9|b^2=g,d^2=g,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8521375);