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) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,15,18)(11,16,19,14)(12,13,17,24)(20,21,22,23)(25,26,30,34)(27,35,36,31)(28,32,38,39)(29,37,40,33);(1,3)(2,6)(4,8)(5,7)(9,11,15,19)(10,16,18,14)(12,22,17,20)(13,23,24,21)(25,27,30,36)(26,31,34,35)(28,33,38,37)(29,32,40,39);(2,4)(3,7)(11,19)(12,17)(13,24)(14,16)(25,27,30,36)(26,31,34,35)(28,37,38,33)(29,39,40,32);(1,3,5,7)(2,6,4,8)(9,12)(10,13)(11,20)(14,23)(15,17)(16,21)(18,24)(19,22)(25,27)(26,31)(28,37)(29,32)(30,36)(33,38)(34,35)(39,40);(1,3)(2,6)(4,8)(5,7)(9,13)(10,17)(11,21)(12,18)(14,20)(15,24)(16,22)(19,23)(25,28,30,38)(26,32,34,39)(27,33,36,37)(29,31,40,35);(1,4,5,2)(3,8,7,6)(9,14)(10,11)(12,23)(13,20)(15,16)(17,21)(18,19)(22,24)(25,29,30,40)(26,33,34,37)(27,32,36,39)(28,31,38,35)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,c^2=g,d^2=gh,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8969349);