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 | |
4289 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(11,17)(14,20)(18,23)(21,24)(26,31)(28,32)(29,37)(34,40);(2,5)(6,7)(9,10,13,16)(11,14,17,20)(12,15,19,22)(18,21,23,24)(25,26,30,31)(27,32,35,28)(29,33,37,38)(34,39,40,36);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,30,35)(26,32,31,28)(29,34,37,40)(33,39,38,36);(1,3,4,8)(2,6,5,7)(9,10)(11,14)(12,15)(13,16)(17,20)(18,21)(19,22)(23,24)(25,28,30,32)(26,27,31,35)(29,36,37,39)(33,34,38,40);(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27)(26,32)(28,31)(29,34)(30,35)(33,39)(36,38)(37,40);(9,12,13,19)(10,15,16,22)(11,18,17,23)(14,21,20,24)(25,29)(26,33)(27,34)(28,36)(30,37)(31,38)(32,39)(35,40)Polycyclic presentation:
2^9|b^2=ghi,c^2=gi,d^2=g,f^2=h,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^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8034792);