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 | |
4715 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(10,15)(11,17)(18,22)(19,24);(3,7)(6,8)(9,10,14,15)(11,16,17,12)(13,18,21,22)(19,23,24,20)(25,26)(27,31)(28,35)(29,30)(32,37)(33,40)(34,38)(36,39);(1,2,5,4)(3,6,7,8)(13,21)(18,22)(19,24)(20,23)(26,30)(31,38)(32,35)(39,40);(1,3,5,7)(2,6,4,8)(9,11)(10,16)(12,15)(13,19)(14,17)(18,23)(20,22)(21,24)(25,27,29,34)(26,31,30,38)(28,36,37,33)(32,40,35,39);(1,4,5,2)(3,8,7,6)(9,12,14,16)(10,17,15,11)(13,20,21,23)(18,24,22,19)(25,28)(26,32)(27,33)(29,37)(30,35)(31,39)(34,36)(38,40);(9,13)(10,18)(11,19)(12,20)(14,21)(15,22)(16,23)(17,24)(25,26)(27,31)(28,32)(29,30)(33,39)(34,38)(35,37)(36,40)Polycyclic presentation:
2^9|b^2=g,c^2=h,d^2=hi,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,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, 8262935);