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 | |
4272 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(9,10,14,13)(11,15,20,18)(12,16,21,22)(17,23,19,24)(25,26,31,36)(27,34,29,32)(28,30,35,33)(37,40,39,38);(3,8)(6,7)(9,11)(10,15)(12,19)(13,18)(14,20)(16,24)(17,21)(22,23)(25,27,31,29)(26,32,36,34)(28,39,35,37)(30,40,33,38);(1,3,5,8)(2,6,4,7)(9,12,14,21)(10,16,13,22)(11,17,20,19)(15,23,18,24)(25,28,31,35)(26,33,36,30)(27,37,29,39)(32,40,34,38);(3,8)(6,7)(9,13,14,10)(11,18,20,15)(12,16,21,22)(17,23,19,24)(25,29,31,27)(26,34,36,32)(28,39,35,37)(30,40,33,38);(1,3)(2,7)(4,6)(5,8)(9,12)(10,16)(11,19)(13,22)(14,21)(15,24)(17,20)(18,23)(25,30,31,33)(26,35,36,28)(27,38,29,40)(32,39,34,37);(1,4,5,2)(3,6,8,7)(9,11)(10,15)(12,19)(13,18)(14,20)(16,24)(17,21)(22,23)(25,26)(27,32)(28,33)(29,34)(30,35)(31,36)(37,40)(38,39)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=i,d^2=ghi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8900827);