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 | |
4483 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)(11,16)(12,17)(13,18)(14,15)(19,23)(20,24)(21,22)(25,26,30,35)(27,33,37,31)(28,38,39,32)(29,34,40,36);(2,5)(6,7)(9,11,15,12)(10,16,14,17)(13,20,22,19)(18,24,21,23)(25,27)(26,31)(28,36)(29,32)(30,37)(33,35)(34,39)(38,40);(1,2)(3,6)(4,5)(7,8)(10,14)(11,12)(18,21)(19,20)(28,39)(29,40)(32,38)(34,36);(1,3,4,8)(2,6,5,7)(9,12,15,11)(10,17,14,16)(13,19,22,20)(18,23,21,24)(25,28,30,39)(26,32,35,38)(27,36,37,34)(29,33,40,31);(9,13,15,22)(10,18,14,21)(11,19,12,20)(16,23,17,24)(25,27)(26,33)(28,34)(29,32)(30,37)(31,35)(36,39)(38,40);(9,14,15,10)(11,17,12,16)(13,21,22,18)(19,24,20,23)(25,29)(26,34)(27,32)(28,33)(30,40)(31,39)(35,36)(37,38)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8815245);