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 | |
4514 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)(5,8)(6,7)(9,10,14,15)(11,13,17,23)(12,16,21,24)(18,22,19,20)(25,26)(27,37)(28,34)(29,30)(31,35)(32,36)(33,40)(38,39);(1,3)(2,6)(4,8)(5,7)(9,11)(10,13)(12,19)(14,17)(15,23)(16,20)(18,21)(22,24)(25,27,32,40)(26,33,36,37)(28,38,29,31)(30,39,34,35);(1,4)(2,7)(3,8)(5,6)(10,15)(11,17)(16,24)(18,19)(25,28,32,29)(26,34,36,30)(27,38,40,31)(33,35,37,39);(2,8)(4,6)(9,11)(10,13)(12,19)(14,17)(15,23)(16,20)(18,21)(22,24)(25,29,32,28)(26,34,36,30)(27,31,40,38)(33,35,37,39);(1,4)(2,3)(5,6)(7,8)(9,12,14,21)(10,16,15,24)(11,18,17,19)(13,22,23,20)(25,30,32,34)(26,29,36,28)(27,39,40,35)(31,37,38,33);(1,3)(2,6)(4,8)(5,7)(9,13)(10,11)(12,20)(14,23)(15,17)(16,19)(18,24)(21,22)(25,31)(26,35)(27,28)(29,40)(30,33)(32,38)(34,37)(36,39)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,c^2=ghi,d^2=ghi,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10049353);