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 | |
4590 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,14)(11,12,16,19)(13,21,22,17)(18,24,23,20)(25,26)(27,28)(29,38)(30,33)(31,34)(32,35)(36,39)(37,40);(1,2,5,6)(3,7,8,4)(9,11)(10,12)(13,18)(14,19)(15,16)(17,20)(21,24)(22,23)(25,27)(26,29)(28,35)(30,36)(31,37)(32,38)(33,34)(39,40);(2,6)(4,7)(10,14)(11,16)(17,21)(18,23)(25,28,32,29)(26,27,35,38)(30,39,37,34)(31,33,36,40);(1,3)(2,4)(5,8)(6,7)(9,12,15,19)(10,16,14,11)(13,20,22,24)(17,23,21,18)(25,29,32,28)(26,27,35,38)(30,34,37,39)(31,33,36,40);(9,13,15,22)(10,17,14,21)(11,18,16,23)(12,20,19,24)(25,30,32,37)(26,33,35,40)(27,36,38,31)(28,39,29,34);(1,4)(2,3)(5,7)(6,8)(9,14,15,10)(11,19,16,12)(13,21,22,17)(18,24,23,20)(25,31,32,36)(26,34,35,39)(27,37,38,30)(28,40,29,33)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=g,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8090111);