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 | #47931 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4262 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,14)(11,17)(16,19)(18,20)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(1,2,7,8)(3,4,6,5)(9,10,13,14)(11,15,17,22)(12,19,21,16)(18,24,20,23)(26,30)(31,33)(32,37)(38,39);(1,3,7,6)(2,5,8,4)(9,11)(10,15)(12,18)(13,17)(14,22)(16,23)(19,24)(20,21)(25,26)(27,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,40);(1,4,7,5)(2,3,8,6)(9,10)(11,15)(12,16)(13,14)(17,22)(18,23)(19,21)(20,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(1,5,7,4)(2,3,8,6)(9,12,13,21)(10,16,14,19)(11,18,17,20)(15,23,22,24);(1,6,7,3)(2,5,8,4)(9,11)(10,15)(12,20)(13,17)(14,22)(16,24)(18,21)(19,23)(25,28,29,36)(26,32,30,37)(27,34,35,40)(31,38,33,39)Polycyclic presentation:
2^9|b^2=gi,c^2=ghi,d^2=ghi,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dghi,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9685670);