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 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4117 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,18)(12,13)(14,15)(16,19)(17,22)(20,23)(21,24)(25,26,31,35)(27,29,33,40)(28,34,30,32)(36,39,37,38);(1,2,5,4)(3,7,8,6)(10,15)(11,19)(12,22)(21,23)(25,27)(26,29)(28,36)(30,37)(31,33)(32,38)(34,39)(35,40);(2,4)(6,7)(10,15)(13,17)(16,18)(21,23)(25,28,31,30)(26,32,35,34)(27,36,33,37)(29,38,40,39);(9,11,14,19)(10,16,15,18)(12,20,22,24)(13,21,17,23)(25,29)(26,33)(27,35)(28,38)(30,39)(31,40)(32,37)(34,36);(1,3,5,8)(2,6,4,7)(9,12)(10,17)(11,20)(13,15)(14,22)(16,23)(18,21)(19,24)(25,30,31,28)(26,34,35,32)(27,37,33,36)(29,39,40,38);(1,4,5,2)(3,7,8,6)(9,13,14,17)(10,12,15,22)(11,21,19,23)(16,20,18,24)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=gh,d^2=g,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8161726);