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 | |
4610 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,19,20,16)(12,13,18,17)(21,22,24,23)(25,26)(27,32)(28,29)(30,35)(31,33)(34,39)(36,40)(37,38);(2,5)(6,8)(10,15)(11,20)(12,18)(22,23)(25,27,31,36)(26,32,33,40)(28,37,34,35)(29,38,39,30);(1,2,4,5)(3,6,7,8)(10,15)(12,18)(16,19)(21,24)(26,33)(27,36)(29,39)(35,37);(3,7)(6,8)(9,11)(10,16)(12,21)(13,23)(14,20)(15,19)(17,22)(18,24)(25,28,31,34)(26,29,33,39)(27,35,36,37)(30,40,38,32);(1,2)(3,6)(4,5)(7,8)(9,12,14,18)(10,17,15,13)(11,21,20,24)(16,22,19,23)(25,29)(26,34)(27,30)(28,33)(31,39)(32,37)(35,40)(36,38);(1,3)(2,6)(4,7)(5,8)(9,13,14,17)(10,18,15,12)(11,22,20,23)(16,21,19,24)(25,30)(26,35)(27,29)(28,32)(31,38)(33,37)(34,40)(36,39)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=ghi,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8506566);