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 | |
4658 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(5,8)(9,10,15,17)(11,12,13,16)(14,22,23,18)(19,24,21,20)(29,37)(32,38)(34,39)(35,40);(1,2,4,6)(3,5,7,8)(9,11,15,13)(10,16,17,12)(14,19,23,21)(18,24,22,20)(25,26)(27,28)(29,32)(30,31)(33,36)(34,35)(37,38)(39,40);(1,3)(2,5)(4,7)(6,8)(10,17)(12,16)(18,22)(20,24)(26,31)(27,33)(32,38)(34,39);(1,2)(3,5)(4,6)(7,8)(9,12,15,16)(10,11,17,13)(14,20,23,24)(18,19,22,21)(25,27,30,33)(26,28,31,36)(29,34,37,39)(32,35,38,40);(1,3)(2,5)(4,7)(6,8)(9,13,15,11)(10,12,17,16)(14,21,23,19)(18,20,22,24)(25,28)(26,27)(29,35)(30,36)(31,33)(32,34)(37,40)(38,39);(9,14)(10,18)(11,19)(12,20)(13,21)(15,23)(16,24)(17,22)(25,29)(26,32)(27,34)(28,35)(30,37)(31,38)(33,39)(36,40)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=gi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8066554);