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 | #50099 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4526 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,13)(11,18,16,12)(14,17,22,21)(19,24,23,20)(25,26)(27,34)(28,33)(29,38)(30,36)(31,32)(35,39)(37,40);(1,2,5,6)(3,7,8,4)(9,11,15,16)(10,12,13,18)(14,19,22,23)(17,20,21,24)(26,32)(29,39)(33,34)(36,37);(2,6)(4,7)(14,22)(17,21)(19,23)(20,24)(25,27,31,28)(26,33,32,34)(29,36,39,37)(30,38,40,35);(2,6)(3,8)(9,12,15,18)(10,16,13,11)(14,20,22,24)(17,23,21,19)(25,28,31,27)(26,34,32,33)(29,36,39,37)(30,38,40,35);(1,3)(2,4)(5,8)(6,7)(9,13,15,10)(11,12,16,18)(14,21,22,17)(19,20,23,24)(25,29,31,39)(26,35,32,38)(27,36,28,37)(30,34,40,33);(1,4)(2,3)(5,7)(6,8)(9,14,15,22)(10,17,13,21)(11,19,16,23)(12,20,18,24)(25,30,31,40)(26,36,32,37)(27,35,28,38)(29,34,39,33)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=h,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=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, 8294946);