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 | |
4667 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,15)(11,19)(17,20)(18,21)(26,31)(27,33)(29,36)(34,37);(1,2)(3,6)(4,7)(5,8)(9,10,14,15)(11,16,19,22)(12,17,23,20)(13,18,24,21)(25,26,30,31)(27,32,33,40)(28,36,38,29)(34,35,37,39);(1,3)(2,5)(4,8)(6,7)(11,19)(13,24)(16,22)(18,21)(25,27,30,33)(26,32,31,40)(28,34,38,37)(29,39,36,35);(3,8)(5,6)(9,11,14,19)(10,16,15,22)(12,21,23,18)(13,20,24,17)(25,27)(26,32)(28,37)(29,35)(30,33)(31,40)(34,38)(36,39);(1,3)(2,6)(4,8)(5,7)(9,12)(10,17)(11,18)(13,22)(14,23)(15,20)(16,24)(19,21)(25,28)(26,29)(27,34)(30,38)(31,36)(32,39)(33,37)(35,40);(1,2)(3,6)(4,7)(5,8)(9,13,14,24)(10,18,15,21)(11,20,19,17)(12,22,23,16)(25,29)(26,28)(27,35)(30,36)(31,38)(32,37)(33,39)(34,40)Polycyclic presentation:
2^9|b^2=g,c^2=h,d^2=gh,f^2=gh,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, 8811730);