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 | |
4597 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,20)(11,21,24,16)(12,14,19,17)(13,18,22,23)(26,31)(27,33)(29,37)(36,40);(1,2,5,6)(3,7,8,4)(9,11,15,24)(10,16,20,21)(12,13,19,22)(14,23,17,18)(25,26)(27,32)(28,33)(29,34)(30,31)(35,40)(36,38)(37,39);(1,2)(3,7)(4,8)(5,6)(12,19)(13,22)(14,17)(18,23)(25,27,30,33)(26,32,31,28)(29,38,37,35)(34,36,39,40);(2,6)(3,8)(9,12,15,19)(10,17,20,14)(11,22,24,13)(16,23,21,18)(25,27)(26,28)(29,35)(30,33)(31,32)(34,36)(37,38)(39,40);(1,3)(2,4)(5,8)(6,7)(9,13,15,22)(10,18,20,23)(11,12,24,19)(14,16,17,21)(25,28)(26,33)(27,31)(29,36)(30,32)(34,38)(35,39)(37,40);(1,4)(2,3)(5,7)(6,8)(9,14,15,17)(10,19,20,12)(11,23,24,18)(13,16,22,21)(25,29,30,37)(26,34,31,39)(27,35,33,38)(28,36,32,40)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=h,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8968671);