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 | |
4605 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(25,26,31,33)(27,35,38,32)(28,29,39,40)(30,34,37,36);(1,2)(3,7)(4,8)(5,6)(9,10,14,15)(11,13,17,22)(12,16,21,23)(18,20,19,24)(25,27,31,38)(26,32,33,35)(28,36,39,34)(29,37,40,30);(1,3,5,4)(2,7,6,8)(10,15)(13,22)(16,23)(20,24)(26,33)(28,39)(32,35)(34,36);(1,4,5,3)(2,8,6,7)(9,11,14,17)(10,13,15,22)(12,19,21,18)(16,24,23,20)(25,28)(26,29)(27,36)(30,35)(31,39)(32,37)(33,40)(34,38);(9,12,14,21)(10,16,15,23)(11,18,17,19)(13,20,22,24)(25,29)(26,28)(27,37)(30,38)(31,40)(32,36)(33,39)(34,35);(9,13)(10,17)(11,15)(12,20)(14,22)(16,19)(18,23)(21,24)(25,30,31,37)(26,34,33,36)(27,29,38,40)(28,35,39,32)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=gh,d^2=ghi,e^2=i,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,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, 7685121);