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 | |
4521 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10,15,19)(11,20,21,16)(12,13,17,23)(14,18,22,24)(25,26,30,33)(27,34,28,31)(29,32,37,40)(35,39,36,38);(1,2)(3,6)(4,7)(5,8)(9,11,15,21)(10,16,19,20)(12,22,17,14)(13,18,23,24);(11,21)(14,22)(16,20)(18,24)(25,27,30,28)(26,31,33,34)(29,36,37,35)(32,39,40,38);(9,12)(10,13)(11,22)(14,21)(15,17)(16,18)(19,23)(20,24)(25,26)(27,31)(28,34)(29,32)(30,33)(35,38)(36,39)(37,40);(1,3,5,4)(2,6,8,7)(9,13)(10,17)(11,18)(12,19)(14,16)(15,23)(20,22)(21,24)(25,28,30,27)(26,31,33,34)(29,36,37,35)(32,38,40,39);(1,4,5,3)(2,7,8,6)(9,14,15,22)(10,18,19,24)(11,12,21,17)(13,16,23,20)(25,29,30,37)(26,32,33,40)(27,35,28,36)(31,38,34,39)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=h,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,e^d=ehi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7559144);