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 | |
4403 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,12)(11,16,18,15)(13,20,21,17)(19,22,24,23)(25,26,31,34)(27,35,38,29)(28,33,30,32)(36,39,37,40);(1,2)(3,8)(4,5)(6,7)(11,18)(13,21)(15,16)(17,20)(25,27,31,38)(26,29,34,35)(28,36,30,37)(32,39,33,40);(2,7)(3,4)(9,11,14,18)(10,15,12,16)(13,19,21,24)(17,22,20,23)(25,27)(26,29)(28,36)(30,37)(31,38)(32,39)(33,40)(34,35);(1,3,6,4)(2,8,7,5)(9,12,14,10)(11,15,18,16)(13,20,21,17)(19,22,24,23)(25,28,31,30)(26,32,34,33)(27,36,38,37)(29,39,35,40);(1,4,6,3)(2,5,7,8)(9,11)(10,16)(12,15)(13,19)(14,18)(17,23)(20,22)(21,24)(25,29)(26,27)(28,39)(30,40)(31,35)(32,36)(33,37)(34,38);(1,5)(2,3)(4,7)(6,8)(9,13)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24)(25,30,31,28)(26,33,34,32)(27,37,38,36)(29,40,35,39)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=gh,d^2=i,e^2=hi,f^2=g,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, 8705716);