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 | |
4666 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)(11,18)(12,16)(13,21)(14,15)(17,23)(19,20)(22,24)(25,26)(27,36)(28,30)(29,34)(31,35)(32,39)(33,37)(38,40);(1,2)(3,8)(4,6)(5,7)(10,15)(12,18)(17,21)(20,22)(25,27,31,39)(26,32,35,36)(28,29,37,40)(30,38,33,34);(1,3)(2,4)(5,6)(7,8)(10,15)(11,16)(17,21)(19,24)(25,28,31,37)(26,30,35,33)(27,29,39,40)(32,38,36,34);(3,6)(4,8)(9,11)(10,12)(13,19)(14,16)(15,18)(17,20)(21,22)(23,24)(25,28)(26,33)(27,29)(30,35)(31,37)(32,34)(36,38)(39,40);(1,4,5,8)(2,6,7,3)(9,12)(10,16)(11,15)(13,22)(14,18)(17,19)(20,23)(21,24)(25,29)(26,34)(27,37)(28,39)(30,32)(31,40)(33,36)(35,38);(1,2)(3,4)(5,7)(6,8)(9,13,14,23)(10,17,15,21)(11,19,16,24)(12,20,18,22)(25,30,31,33)(26,28,35,37)(27,38,39,34)(29,36,40,32)Polycyclic presentation:
2^9|b^2=gh,c^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9068110);