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 | |
4582 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,8,6,4)(9,10)(11,17)(12,13)(14,18)(15,16)(19,24)(20,21)(22,23)(26,30)(27,33)(32,35)(34,39);(1,3)(2,4)(5,6)(7,8)(10,16)(12,17)(18,23)(20,24)(26,30)(31,37)(32,35)(38,40);(1,4,5,8)(2,6,7,3)(9,11,15,13)(10,17,16,12)(14,21,22,19)(18,20,23,24)(25,26)(27,31)(28,35)(29,30)(32,36)(33,37)(34,40)(38,39);(1,3)(2,4)(5,6)(7,8)(9,12)(10,11)(13,16)(14,20)(15,17)(18,19)(21,23)(22,24);(1,3)(2,4)(5,6)(7,8)(9,13,15,11)(10,12,16,17)(14,21,22,19)(18,20,23,24)(25,27,29,33)(26,31,30,37)(28,34,36,39)(32,38,35,40);(1,4)(2,3)(5,8)(6,7)(9,14,15,22)(10,18,16,23)(11,19,13,21)(12,20,17,24)(25,28,29,36)(26,32,30,35)(27,34,33,39)(31,38,37,40)Polycyclic presentation:
2^9|a^2=hi,c^2=g,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, 8087601);