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 | |
4211 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)(11,14)(12,20)(13,16)(15,21)(17,19)(18,23)(22,24)(25,26)(27,29)(28,37)(30,38)(31,33)(32,36)(34,35)(39,40);(1,2,5,4)(3,7,8,6)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,21)(22,24)(25,27,31,36)(26,32,33,29)(28,34,38,40)(30,35,37,39);(1,3,5,8)(2,6,4,7)(9,10,13,16)(11,17,19,14)(12,20,21,15)(18,22,24,23)(25,28,31,38)(26,30,33,37)(27,34,36,40)(29,39,32,35);(1,3,5,8)(2,7,4,6)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,21)(22,24)(25,28)(26,30)(27,34)(29,39)(31,38)(32,35)(33,37)(36,40);(1,3)(2,7)(4,6)(5,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,29,31,32)(26,27,33,36)(28,35,38,39)(30,40,37,34);(1,4,5,2)(3,6,8,7)(9,12)(10,15)(11,18)(13,21)(14,22)(16,20)(17,23)(19,24)(25,30)(26,28)(27,35)(29,40)(31,37)(32,34)(33,38)(36,39)Polycyclic presentation:
2^9|b^2=hi,c^2=gi,d^2=ghi,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9515026);