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 | |
4696 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)(15,17)(16,20)(22,23)(25,26)(27,32)(28,29)(30,35)(31,33)(34,39)(36,40)(37,38);(2,5)(6,8)(9,10)(11,17)(12,16)(13,14)(15,18)(19,23)(20,21)(22,24)(25,27,31,36)(26,32,33,40)(28,37,34,35)(29,38,39,30);(1,2,4,5)(3,6,7,8)(11,18)(15,17)(19,24)(22,23)(26,33)(27,36)(29,39)(35,37);(3,7)(6,8)(9,10)(11,15)(12,20)(13,14)(16,21)(17,18)(19,23)(22,24)(25,28,31,34)(26,29,33,39)(27,35,36,37)(30,40,38,32);(1,2)(3,6)(4,5)(7,8)(9,11,13,18)(10,15,14,17)(12,19,21,24)(16,22,20,23)(25,29)(26,34)(27,30)(28,33)(31,39)(32,37)(35,40)(36,38);(1,3)(2,6)(4,7)(5,8)(9,12,13,21)(10,16,14,20)(11,19,18,24)(15,22,17,23)(25,30)(26,35)(27,29)(28,32)(31,38)(33,37)(34,40)(36,39)Polycyclic presentation:
2^9|b^2=h,c^2=hi,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8506551);