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 | |
4295 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,18)(11,13,16,12)(14,17,23,22)(19,21,24,20)(25,26,32,35)(27,28,37,33)(29,39,38,31)(30,40,36,34);(1,2)(3,8)(4,6)(5,7)(9,11,15,16)(10,12,18,13)(14,19,23,24)(17,20,22,21)(25,27,32,37)(26,33,35,28)(29,40,38,34)(30,31,36,39);(1,3,5,6)(2,4,7,8)(9,12,15,13)(10,11,18,16)(14,20,23,21)(17,19,22,24)(25,28)(26,27)(29,36)(30,38)(31,34)(32,33)(35,37)(39,40);(1,2)(3,4)(5,7)(6,8)(9,10)(11,12)(13,16)(14,22)(15,18)(17,23)(19,21)(20,24)(25,29,32,38)(26,31,35,39)(27,34,37,40)(28,36,33,30);(9,13,15,12)(10,16,18,11)(14,21,23,20)(17,24,22,19)(25,30,32,36)(26,34,35,40)(27,31,37,39)(28,38,33,29);(1,4)(2,6)(3,7)(5,8)(9,14,15,23)(10,17,18,22)(11,19,16,24)(12,20,13,21)(25,31)(26,29)(27,36)(28,34)(30,37)(32,39)(33,40)(35,38)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=ghi,d^2=h,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8332085);