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:
(10,15)(11,17)(18,22)(19,24)(25,26)(27,35)(28,33)(29,39)(30,31)(32,37)(34,40)(36,38);(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,24)(21,22)(26,31)(28,38)(32,35)(34,39);(2,5)(3,7)(11,17)(12,16)(13,21)(18,22)(25,27,30,37)(26,32,31,35)(28,39,38,34)(29,33,40,36);(1,3,4,7)(2,6,5,8)(9,11)(10,16)(12,15)(13,19)(14,17)(18,23)(20,22)(21,24)(25,27)(26,32)(28,39)(29,36)(30,37)(31,35)(33,40)(34,38);(1,3)(2,6)(4,7)(5,8)(9,12,14,16)(10,17,15,11)(13,20,21,23)(18,24,22,19)(25,28,30,38)(26,33,31,36)(27,34,37,39)(29,35,40,32);(1,2)(3,6)(4,5)(7,8)(9,13)(10,18)(11,19)(12,20)(14,21)(15,22)(16,23)(17,24)(25,29)(26,34)(27,36)(28,35)(30,40)(31,39)(32,38)(33,37)Polycyclic presentation:
2^9|c^2=h,d^2=i,e^2=g,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, 8891048);