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 | |
4537 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)(11,15)(12,17)(13,21)(14,16)(18,23)(19,22)(20,24)(25,26,31,35)(27,28,38,32)(29,33,37,40)(30,34,36,39);(1,2,5,4)(3,7,8,6)(9,11,14,19)(10,15,16,22)(12,18,20,21)(13,17,23,24)(25,27)(26,32)(28,35)(29,36)(30,37)(31,38)(33,34)(39,40);(1,3,5,8)(2,6,4,7)(10,16)(11,19)(12,20)(13,23)(29,37)(30,36)(33,40)(34,39);(1,3,5,8)(2,7,4,6)(9,12)(10,17)(11,18)(13,22)(14,20)(15,23)(16,24)(19,21)(25,28,31,32)(26,27,35,38)(29,34,37,39)(30,40,36,33);(1,3)(2,7)(4,6)(5,8)(9,11)(10,15)(12,21)(13,17)(14,19)(16,22)(18,20)(23,24)(25,29,31,37)(26,33,35,40)(27,36,38,30)(28,39,32,34);(1,4,5,2)(3,6,8,7)(9,13)(10,18)(11,17)(12,22)(14,23)(15,20)(16,21)(19,24)(25,30,31,36)(26,34,35,39)(27,37,38,29)(28,40,32,33)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=hi,d^2=ghi,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9515151);