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 | |
4578 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,16)(11,15,17,24)(12,13,21,23)(18,19,20,22)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,38)(37,40);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,20)(13,22)(14,17)(16,24)(18,21)(19,23)(25,26,29,32)(27,30,35,33)(28,31,36,38)(34,37,40,39);(1,2,4,6)(3,5,8,7)(10,16)(11,17)(12,21)(19,22)(28,36)(31,38)(34,40)(37,39);(9,11)(10,15)(12,18)(13,19)(14,17)(16,24)(20,21)(22,23)(25,27)(26,30)(28,34)(29,35)(31,37)(32,33)(36,40)(38,39);(1,2)(3,5)(4,6)(7,8)(9,12)(10,13)(11,18)(14,21)(15,19)(16,23)(17,20)(22,24)(25,28,29,36)(26,31,32,38)(27,34,35,40)(30,37,33,39);(1,3)(2,5)(4,8)(6,7)(9,13)(10,12)(11,19)(14,23)(15,18)(16,21)(17,22)(20,24)(25,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,38)(39,40)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=hi,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9152556);