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 | |
4528 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10,15,17)(11,21,23,16)(12,24,22,18)(13,19,20,14)(25,26)(27,29)(28,32)(30,34)(31,36)(33,38)(35,39)(37,40);(1,3,5,8)(2,6,7,4)(9,11,15,23)(10,16,17,21)(12,20,22,13)(14,24,19,18)(25,27,30,36)(26,31,34,29)(28,35,38,40)(32,37,33,39);(1,3)(2,4)(5,8)(6,7)(10,17)(12,22)(13,20)(16,21)(25,28,30,38)(26,32,34,33)(27,35,36,40)(29,39,31,37);(1,4)(2,3)(5,6)(7,8)(9,12,15,22)(10,18,17,24)(11,20,23,13)(14,21,19,16)(25,28)(26,33)(27,35)(29,37)(30,38)(31,39)(32,34)(36,40);(9,13,15,20)(10,19,17,14)(11,22,23,12)(16,24,21,18)(25,29)(26,27)(28,37)(30,31)(32,40)(33,35)(34,36)(38,39);(9,14,15,19)(10,20,17,13)(11,18,23,24)(12,21,22,16)(25,27)(26,29)(28,35)(30,36)(31,34)(32,39)(33,37)(38,40)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=ghi,d^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8680075);