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 | |
4647 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(9,10)(11,12)(13,21)(14,15)(16,19)(17,22)(18,23)(20,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,38)(36,40)(37,39);(3,7)(6,8)(10,15)(12,16)(17,21)(20,23)(25,27,31,35)(26,32,30,38)(28,36,37,29)(33,40,39,34);(1,2,5,4)(3,6,7,8)(10,15)(12,16)(13,22)(18,24)(26,30)(27,35)(29,36)(33,39);(9,11,14,19)(10,16,15,12)(13,18,22,24)(17,23,21,20)(25,28)(26,33)(27,36)(29,35)(30,39)(31,37)(32,40)(34,38);(1,3,5,7)(2,6,4,8)(9,12,14,16)(10,11,15,19)(13,20,22,23)(17,18,21,24)(25,29,31,36)(26,34,30,40)(27,37,35,28)(32,39,38,33);(1,4,5,2)(3,8,7,6)(9,13)(10,17)(11,18)(12,20)(14,22)(15,21)(16,23)(19,24)(25,30,31,26)(27,38,35,32)(28,39,37,33)(29,40,36,34)Polycyclic presentation:
2^9|b^2=i,c^2=ghi,d^2=g,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8853394);