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 |
| prime radical | #50642 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4395 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,8,7)(9,10)(11,17)(12,22)(13,18)(14,19)(15,16)(20,21)(23,24)(26,31)(27,34)(28,36)(37,39);(2,6)(4,7)(10,16)(13,19)(17,22)(20,23)(25,26,30,31)(27,29,34,32)(28,33,36,38)(35,37,40,39);(1,3,5,8)(2,7,6,4)(9,11,15,12)(10,17,16,22)(13,20,19,23)(14,24,18,21)(25,27,30,34)(26,32,31,29)(28,35,36,40)(33,39,38,37);(1,4)(2,3)(5,7)(6,8)(9,12,15,11)(10,17,16,22)(13,23,19,20)(14,24,18,21)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(1,4)(2,3)(5,7)(6,8)(9,13,15,19)(10,18,16,14)(11,20,12,23)(17,21,22,24)(25,28,30,36)(26,33,31,38)(27,35,34,40)(29,37,32,39);(1,3)(2,4)(5,8)(6,7)(9,14,15,18)(10,19,16,13)(11,21,12,24)(17,20,22,23)(25,29,30,32)(26,34,31,27)(28,37,36,39)(33,40,38,35)Polycyclic presentation:
2^9|a^2=hi,b^2=i,c^2=g,d^2=gh,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8163821);