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 | |
4555 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,7,8)(3,5,6,4)(9,10)(11,15)(12,17)(13,21)(14,16)(18,22)(19,20)(23,24)(25,26)(27,38)(28,31)(29,36)(30,35)(32,37)(33,40)(34,39);(1,3,7,6)(2,4,8,5)(9,11,14,17)(10,15,16,12)(13,19,22,24)(18,23,21,20)(25,27)(26,33)(28,29)(30,39)(31,35)(32,40)(34,36)(37,38);(1,4,7,5)(2,3,8,6)(10,16)(11,17)(18,21)(19,24)(25,28,32,39)(26,34,37,31)(27,29,40,30)(33,36,38,35);(1,5,7,4)(2,3,8,6)(9,12,14,15)(10,17,16,11)(13,20,22,23)(18,24,21,19)(25,29,32,30)(26,35,37,36)(27,28,40,39)(31,38,34,33);(9,13,14,22)(10,18,16,21)(11,19,17,24)(12,20,15,23)(25,30,32,29)(26,36,37,35)(27,39,40,28)(31,33,34,38);(1,6,7,3)(2,4,8,5)(9,10)(11,15)(12,17)(13,18)(14,16)(19,23)(20,24)(21,22)(25,31,32,34)(26,28,37,39)(27,35,40,36)(29,38,30,33)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=g,d^2=gi,e^2=h,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, 8094054);