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 | #39548 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4524 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(3,4)(9,10)(11,14)(12,17)(13,18)(15,19)(16,21)(20,22)(23,24)(25,26,30,32)(27,31,34,39)(28,37,33,29)(35,40,38,36);(1,2,6,5)(3,8,4,7)(10,11)(15,17)(16,18)(22,23)(25,27,30,34)(26,31,32,39)(28,35,33,38)(29,36,37,40);(1,3,6,4)(2,7,5,8)(10,11)(12,19)(16,18)(20,24)(26,32)(27,34)(28,33)(36,40);(1,4,6,3)(2,8,5,7)(9,11,14,10)(12,17,19,15)(13,18,21,16)(20,23,24,22)(25,27)(26,31)(28,35)(29,36)(30,34)(32,39)(33,38)(37,40);(1,5,6,2)(3,8,4,7)(9,12,14,19)(10,15,11,17)(13,20,21,24)(16,22,18,23)(25,28,30,33)(26,29,32,37)(27,35,34,38)(31,36,39,40);(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,29)(26,33)(27,36)(28,32)(30,37)(31,38)(34,40)(35,39)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=ghi,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8034633);