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 | |
4457 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,18)(11,16,20,14)(12,21,13,17)(19,24,22,23)(25,26,31,29)(27,34,32,28)(30,37,39,33)(35,36,40,38);(1,2)(3,5)(4,6)(7,8)(9,11)(10,16)(12,22)(13,19)(14,18)(15,20)(17,24)(21,23)(25,27)(26,28)(29,34)(30,35)(31,32)(33,36)(37,38)(39,40);(1,2,4,6)(3,7,8,5)(9,12,15,13)(10,17,18,21)(11,19,20,22)(14,24,16,23)(25,28,31,34)(26,32,29,27)(30,38,39,36)(33,35,37,40);(3,8)(5,7)(9,13,15,12)(10,17,18,21)(11,19,20,22)(14,23,16,24)(25,27)(26,28)(29,34)(30,35)(31,32)(33,36)(37,38)(39,40);(1,3)(2,5)(4,8)(6,7)(9,14)(10,11)(12,23)(13,24)(15,16)(17,22)(18,20)(19,21)(25,29,31,26)(27,28,32,34)(30,37,39,33)(35,36,40,38);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,22)(13,19)(15,20)(16,18)(17,23)(21,24)(25,30,31,39)(26,33,29,37)(27,35,32,40)(28,36,34,38)Polycyclic presentation:
2^9|a^2=h,c^2=gi,d^2=gh,e^2=g,f^2=g,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, 8850184);