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 | |
4655 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,5)(3,4,8,7)(9,10,14,17)(11,15,18,22)(12,13,16,21)(19,20,23,24)(25,26,30,31)(27,32,35,39)(28,29,38,36)(33,40,37,34);(2,5)(3,8)(9,11)(10,15)(12,19)(13,20)(14,18)(16,23)(17,22)(21,24)(25,26,30,31)(27,32,35,39)(28,36,38,29)(33,34,37,40);(1,3,6,8)(2,7,5,4)(11,18)(15,22)(19,23)(20,24)(26,31)(29,36)(32,39)(34,40);(1,4)(2,3)(5,8)(6,7)(9,12)(10,13)(11,19)(14,16)(15,20)(17,21)(18,23)(22,24)(25,27,30,35)(26,32,31,39)(28,37,38,33)(29,34,36,40);(1,5,6,2)(3,4,8,7)(9,13)(10,16)(11,20)(12,17)(14,21)(15,23)(18,24)(19,22)(25,28)(26,29)(27,33)(30,38)(31,36)(32,40)(34,39)(35,37);(1,4)(2,3)(5,8)(6,7)(9,11)(10,15)(12,19)(13,20)(14,18)(16,23)(17,22)(21,24)(25,29)(26,28)(27,34)(30,36)(31,38)(32,37)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=g,d^2=gh,e^2=g,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, 8800857);