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 | |
4586 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(25,26,32,31)(27,36,30,33)(28,35,29,34)(37,39,38,40);(1,3)(2,4)(5,7)(6,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,27,32,30)(26,33,31,36)(28,38,29,37)(34,40,35,39);(2,6)(3,7)(10,14)(15,17)(16,21)(22,23)(25,28,32,29)(26,34,31,35)(27,37,30,38)(33,39,36,40);(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,29,32,28)(26,35,31,34)(27,38,30,37)(33,40,36,39);(1,4)(2,3)(5,8)(6,7)(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(25,30,32,27)(26,36,31,33)(28,38,29,37)(34,40,35,39);(1,3)(2,4)(5,7)(6,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,31,32,26)(27,36,30,33)(28,35,29,34)(37,40,38,39)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,d^2=ghi,e^2=ghi,f^2=g,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, 8100899);