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 | |
4665 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)(9,10,15,18)(11,16,21,22)(12,20,23,17)(13,19,24,14)(25,26,32,36)(27,33,38,28)(29,40,37,34)(30,35,39,31);(1,2,5,6)(3,7,4,8)(9,11)(10,16)(12,14)(13,20)(15,21)(17,24)(18,22)(19,23)(25,27)(26,33)(28,36)(29,35)(30,34)(31,37)(32,38)(39,40);(9,12,15,23)(10,17,18,20)(11,19,21,14)(13,22,24,16)(25,28)(26,27)(29,39)(30,37)(31,40)(32,33)(34,35)(36,38);(1,3,5,4)(2,7,6,8)(9,12)(10,17)(11,14)(13,22)(15,23)(16,24)(18,20)(19,21)(25,29,32,37)(26,34,36,40)(27,35,38,31)(28,39,33,30);(9,13,15,24)(10,14,18,19)(11,20,21,17)(12,22,23,16)(25,30,32,39)(26,35,36,31)(27,34,38,40)(28,37,33,29);(1,4,5,3)(2,8,6,7)(9,14)(10,13)(11,12)(15,19)(16,20)(17,22)(18,24)(21,23)(25,31,32,35)(26,30,36,39)(27,37,38,29)(28,40,33,34)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=h,d^2=ghi,e^2=i,f^2=ghi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7804955);