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 | |
4492 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,7,8)(3,5,6,4)(25,26)(27,30)(28,33)(29,32)(31,36)(34,37)(35,39)(38,40);(1,3,7,6)(2,4,8,5)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(26,27)(31,33)(32,34)(38,39);(1,4,7,5)(2,3,8,6)(10,14)(15,17)(16,21)(22,23)(25,26,30,27)(28,31,36,33)(29,32,37,34)(35,38,40,39);(1,5,7,4)(2,3,8,6)(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,27,30,26)(28,33,36,31)(29,34,37,32)(35,39,40,38);(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(25,28)(26,31)(27,33)(29,35)(30,36)(32,38)(34,39)(37,40);(1,6,7,3)(2,4,8,5)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,29,30,37)(26,32,27,34)(28,35,36,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=g,d^2=ghi,e^2=hi,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, 8100939);