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 | |
4546 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)(9,10,15,16)(11,17,12,14)(13,21,23,18)(19,22,20,24)(26,29)(28,32)(33,37)(36,38);(3,7)(6,8)(10,16)(11,12)(18,21)(19,20)(25,26,31,29)(27,32,35,28)(30,33,39,37)(34,38,40,36);(1,2,5,4)(3,6,7,8)(9,11,15,12)(10,14,16,17)(13,19,23,20)(18,22,21,24)(25,27,31,35)(26,32,29,28)(30,34,39,40)(33,38,37,36);(1,3,5,7)(2,6,4,8)(9,12,15,11)(10,17,16,14)(13,20,23,19)(18,24,21,22)(25,28)(26,27)(29,35)(30,38)(31,32)(33,40)(34,37)(36,39);(1,4,5,2)(3,8,7,6)(9,13,15,23)(10,18,16,21)(11,19,12,20)(14,22,17,24)(25,29,31,26)(27,28,35,32)(30,37,39,33)(34,36,40,38);(9,14,15,17)(10,11,16,12)(13,22,23,24)(18,19,21,20)(25,30)(26,33)(27,34)(28,36)(29,37)(31,39)(32,38)(35,40)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=ghi,d^2=h,e^2=ghi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8374375);