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 | |
4366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,19)(12,14)(13,16)(15,17)(18,23)(20,21)(22,24)(25,26,31,33)(27,32,35,29)(28,34,38,40)(30,36,37,39);(1,2,5,4)(3,6,7,8)(9,11,15,13)(10,16,17,19)(12,22,23,20)(14,21,18,24)(25,27,31,35)(26,32,33,29)(28,37,38,30)(34,39,40,36);(2,4)(6,8)(10,17)(14,18)(16,19)(21,24)(26,33)(27,35)(30,37)(34,40);(3,7)(6,8)(9,12)(10,18)(11,20)(13,22)(14,17)(15,23)(16,21)(19,24)(25,28)(26,34)(27,30)(29,39)(31,38)(32,36)(33,40)(35,37);(1,3)(2,6)(4,8)(5,7)(9,13,15,11)(10,19,17,16)(12,20,23,22)(14,24,18,21)(25,29,31,32)(26,35,33,27)(28,36,38,39)(30,40,37,34);(1,4,5,2)(3,8,7,6)(9,14,15,18)(10,12,17,23)(11,21,13,24)(16,22,19,20)(25,30)(26,36)(27,28)(29,40)(31,37)(32,34)(33,39)(35,38)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8233054);