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 | |
4672 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(10,15)(11,17)(16,23)(18,20)(25,26,30,34)(27,35,36,31)(28,32,38,39)(29,33,37,40);(1,3,4,7)(2,5,6,8)(9,10)(11,13)(12,16)(14,15)(17,22)(18,21)(19,23)(20,24)(27,36)(29,37)(31,35)(33,40);(1,2)(3,5)(4,6)(7,8)(11,17)(12,19)(13,22)(16,23)(25,27,30,36)(26,31,34,35)(28,29,38,37)(32,40,39,33);(9,11)(10,13)(12,20)(14,17)(15,22)(16,24)(18,19)(21,23)(25,28,30,38)(26,32,34,39)(27,37,36,29)(31,33,35,40);(1,3)(2,5)(4,7)(6,8)(9,12)(10,16)(11,18)(13,21)(14,19)(15,23)(17,20)(22,24)(25,29,30,37)(26,33,34,40)(27,28,36,38)(31,39,35,32);(1,2)(3,5)(4,6)(7,8)(9,13)(10,11)(12,21)(14,22)(15,17)(16,18)(19,24)(20,23)(25,26)(27,35)(28,32)(29,33)(30,34)(31,36)(37,40)(38,39)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=i,d^2=i,e^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8922884);