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 | |
4439 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,17)(11,18,13,16)(12,21,23,14)(19,20,22,24)(25,26,31,33)(27,37,38,32)(28,29,35,34)(30,40,39,36);(1,2)(3,8)(4,6)(5,7)(9,11,15,13)(10,16,17,18)(12,19,23,22)(14,20,21,24)(25,27)(26,32)(28,39)(29,40)(30,35)(31,38)(33,37)(34,36);(1,3,5,6)(2,4,7,8)(26,33)(27,38)(28,35)(36,40);(9,12)(10,14)(11,19)(13,22)(15,23)(16,20)(17,21)(18,24)(25,28,31,35)(26,34,33,29)(27,39,38,30)(32,36,37,40);(1,2)(3,4)(5,7)(6,8)(9,13,15,11)(10,16,17,18)(12,22,23,19)(14,20,21,24)(25,29,31,34)(26,35,33,28)(27,36,38,40)(30,32,39,37);(1,4)(2,6)(3,7)(5,8)(9,14)(10,12)(11,20)(13,24)(15,21)(16,19)(17,23)(18,22)(25,30,31,39)(26,36,33,40)(27,35,38,28)(29,32,34,37)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=hi,d^2=h,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9590933);