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 | |
4530 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)(11,16)(12,18)(13,22)(14,20)(15,21)(17,23)(19,24)(25,26,30,31)(27,29,34,32)(28,36,38,33)(35,39,40,37);(1,2)(3,7)(4,6)(5,8)(9,11,15,14)(10,16,21,20)(12,19,23,22)(13,18,24,17)(26,31)(27,34)(33,36)(35,40);(1,3)(2,5)(4,8)(6,7)(9,12)(10,17)(11,22)(13,20)(14,19)(15,23)(16,24)(18,21)(25,27,30,34)(26,32,31,29)(28,35,38,40)(33,39,36,37);(2,6)(5,7)(9,12)(10,18)(11,19)(13,20)(14,22)(15,23)(16,24)(17,21)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(1,3)(2,7)(4,8)(5,6)(9,13)(10,19)(11,18)(12,20)(14,17)(15,24)(16,23)(21,22)(25,28)(26,33)(27,35)(29,37)(30,38)(31,36)(32,39)(34,40);(1,2)(3,5)(4,6)(7,8)(9,14,15,11)(10,20,21,16)(12,19,23,22)(13,18,24,17)(25,29,30,32)(26,34,31,27)(28,37,38,39)(33,40,36,35)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8597172);