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 | |
4573 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)(11,17)(18,23)(19,21)(25,26)(27,35)(28,32)(29,30)(31,34)(33,40)(36,39)(37,38);(1,2)(3,8)(4,6)(5,7)(9,10)(11,16)(12,17)(13,18)(14,15)(19,24)(20,21)(22,23)(27,28)(32,35)(36,38)(37,39);(1,3,5,6)(2,4,7,8)(25,27,31,28)(26,32,34,35)(29,36,40,38)(30,37,33,39);(1,2)(3,4)(5,7)(6,8)(9,11,14,17)(10,16,15,12)(13,21,22,19)(18,20,23,24)(25,28,31,27)(26,32,34,35)(29,38,40,36)(30,37,33,39);(9,12,14,16)(10,17,15,11)(13,20,22,24)(18,21,23,19)(25,29)(26,33)(27,36)(28,38)(30,34)(31,40)(32,39)(35,37);(1,4)(2,6)(3,7)(5,8)(9,13)(10,18)(11,19)(12,20)(14,22)(15,23)(16,24)(17,21)(25,30,31,33)(26,29,34,40)(27,37,28,39)(32,36,35,38)Polycyclic presentation:
2^9|c^2=i,d^2=gh,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8331998);