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 | |
4609 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)(11,12)(13,21)(14,15)(16,23)(17,20)(18,24)(19,22)(25,26)(27,33)(28,36)(29,30)(31,32)(34,38)(35,37)(39,40);(1,2)(3,4)(5,6)(7,8)(9,11,14,17)(10,12,15,20)(13,22,23,18)(16,24,21,19)(26,32)(29,39)(33,36)(34,37);(2,6)(4,7)(10,15)(11,17)(16,21)(18,22)(25,27,31,28)(26,33,32,36)(29,34,39,37)(30,38,40,35);(9,12)(10,11)(13,19)(14,20)(15,17)(16,18)(21,22)(23,24)(25,28,31,27)(26,33,32,36)(29,37,39,34)(30,38,40,35);(1,3)(2,7)(4,6)(5,8)(9,13)(10,16)(11,18)(12,19)(14,23)(15,21)(17,22)(20,24)(25,29)(26,30)(27,34)(28,37)(31,39)(32,40)(33,38)(35,36);(1,4,5,7)(2,3,6,8)(9,10)(11,12)(13,16)(14,15)(17,20)(18,19)(21,23)(22,24)(25,30)(26,29)(27,35)(28,38)(31,40)(32,39)(33,37)(34,36)Polycyclic presentation:
2^9|b^2=h,c^2=g,d^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8139581);