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 | |
4494 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,14)(15,18)(16,20)(22,24)(25,26)(27,30)(28,33)(29,34)(31,32)(35,40)(36,39)(37,38);(1,2)(3,4)(5,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,27,31,35)(26,30,32,40)(28,36,37,29)(33,39,38,34);(2,6)(3,7)(11,17)(15,18)(19,23)(22,24)(26,32)(27,35)(29,36)(33,38);(3,7)(4,8)(9,10)(11,18)(12,20)(13,14)(15,17)(16,21)(19,22)(23,24)(25,28)(26,33)(27,36)(29,35)(30,39)(31,37)(32,38)(34,40);(1,3)(2,4)(5,7)(6,8)(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,24)(21,23)(25,29,31,36)(26,34,32,39)(27,37,35,28)(30,38,40,33);(1,4)(2,3)(5,8)(6,7)(9,12,13,21)(10,16,14,20)(11,19,17,23)(15,22,18,24)(25,30)(26,35)(27,32)(28,34)(29,38)(31,40)(33,36)(37,39)Polycyclic presentation:
2^9|b^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8971554);