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 | |
4416 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)(11,16)(12,18)(13,19)(14,21)(15,17)(20,24)(22,23)(25,26)(27,35)(28,39)(29,34)(30,31)(32,38)(33,37)(36,40);(1,2)(3,5)(4,6)(7,8)(9,11)(10,16)(12,21)(13,15)(14,18)(17,19)(20,22)(23,24)(25,27,31,38)(26,32,30,35)(28,36,37,29)(33,40,39,34);(1,2,4,6)(3,7,8,5)(10,17)(11,13)(12,22)(14,24)(26,30)(28,37)(29,36)(32,35);(3,8)(5,7)(9,12,15,22)(10,18,17,23)(11,21,13,20)(14,19,24,16)(25,28)(26,33)(27,36)(29,38)(30,39)(31,37)(32,40)(34,35);(1,3,4,8)(2,5,6,7)(9,13,15,11)(10,19,17,16)(12,21,22,20)(14,23,24,18)(25,29,31,36)(26,34,30,40)(27,37,38,28)(32,39,35,33);(1,2)(3,5)(4,6)(7,8)(9,14)(10,20)(11,18)(12,19)(13,23)(15,24)(16,22)(17,21)(25,30,31,26)(27,35,38,32)(28,39,37,33)(29,40,36,34)Polycyclic presentation:
2^9|b^2=g,c^2=ghi,d^2=gh,e^2=gi,f^2=g,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, 9430748);