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 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4194 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,14)(12,19)(13,15)(16,21)(17,23)(18,24)(20,22)(25,26,31,32)(27,35,38,29)(28,37,40,33)(30,36,39,34);(1,2,5,3)(4,7,8,6)(9,11,13,17)(10,14,15,23)(12,20,21,18)(16,24,19,22)(26,32)(29,35)(30,39)(33,37);(2,3)(6,7)(10,15)(11,17)(16,19)(18,20)(25,27,31,38)(26,29,32,35)(28,34,40,36)(30,37,39,33);(1,2,5,3)(4,6,8,7)(9,10)(11,14)(12,16)(13,15)(17,23)(18,22)(19,21)(20,24)(25,28,31,40)(26,33,32,37)(27,36,38,34)(29,39,35,30);(1,3,5,2)(4,7,8,6)(9,11)(10,14)(12,18)(13,17)(15,23)(16,22)(19,24)(20,21)(25,29)(26,27)(28,39)(30,40)(31,35)(32,38)(33,36)(34,37);(1,4,5,8)(2,6,3,7)(9,12,13,21)(10,16,15,19)(11,18,17,20)(14,22,23,24)(25,30,31,39)(26,34,32,36)(27,37,38,33)(28,29,40,35)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=g,d^2=hi,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8731012);