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 | |
4457 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,16)(11,17,18,13)(12,15,20,23)(19,24,21,22)(25,26,32,36)(27,31,29,34)(28,38,40,33)(30,35,37,39);(1,2,5,4)(3,7,8,6)(9,10)(11,13)(12,15)(14,16)(17,18)(19,22)(20,23)(21,24)(25,27,32,29)(26,31,36,34)(28,30,40,37)(33,39,38,35);(1,3,5,8)(2,6,4,7)(11,18)(12,20)(13,17)(15,23)(25,28,32,40)(26,33,36,38)(27,37,29,30)(31,35,34,39);(3,8)(6,7)(9,11)(10,13)(12,21)(14,18)(15,24)(16,17)(19,20)(22,23)(25,29,32,27)(26,34,36,31)(28,37,40,30)(33,35,38,39);(1,3)(2,7)(4,6)(5,8)(9,12)(10,15)(11,19)(13,22)(14,20)(16,23)(17,24)(18,21)(25,30,32,37)(26,35,36,39)(27,28,29,40)(31,38,34,33);(1,4,5,2)(3,6,8,7)(9,13)(10,11)(12,22)(14,17)(15,19)(16,18)(20,24)(21,23)(25,31,32,34)(26,27,36,29)(28,39,40,35)(30,38,37,33)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=i,d^2=h,e^2=h,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8935809);