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 | |
4632 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,14,15)(11,12,17,20)(13,16,21,22)(18,19,23,24)(25,26,31,32)(27,34,28,29)(30,33,38,39)(35,40,36,37);(1,2)(3,5)(4,6)(7,8)(25,27,31,28)(26,29,32,34)(30,36,38,35)(33,40,39,37);(10,15)(11,17)(16,22)(18,23)(26,32)(27,28)(30,38)(37,40);(1,3,4,7)(2,5,6,8)(9,11)(10,12)(13,18)(14,17)(15,20)(16,19)(21,23)(22,24)(25,28,31,27)(26,29,32,34)(30,36,38,35)(33,37,39,40);(9,12)(10,11)(13,19)(14,20)(15,17)(16,18)(21,24)(22,23)(25,29)(26,28)(27,32)(30,37)(31,34)(33,36)(35,39)(38,40);(9,13,14,21)(10,16,15,22)(11,18,17,23)(12,19,20,24)(25,30,31,38)(26,33,32,39)(27,35,28,36)(29,37,34,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,d^2=g,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7731530);