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 | |
4508 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(9,10,14,15)(11,16,17,12)(13,21,22,18)(19,20,24,23)(25,26)(27,31)(28,37)(29,34)(30,35)(32,36)(33,39)(38,40);(2,5)(6,7)(10,15)(11,17)(18,21)(19,24)(25,27,30,36)(26,31,35,32)(28,38,39,34)(29,37,40,33);(1,2)(3,6)(4,5)(7,8)(10,15)(11,17)(13,22)(20,23)(27,36)(29,40)(31,32)(34,38);(1,3,4,8)(2,6,5,7)(9,11)(10,16)(12,15)(13,19)(14,17)(18,23)(20,21)(22,24)(25,27)(26,32)(28,38)(29,37)(30,36)(31,35)(33,40)(34,39);(9,12,14,16)(10,17,15,11)(13,20,22,23)(18,24,21,19)(25,28,30,39)(26,33,35,37)(27,34,36,38)(29,31,40,32);(9,13,14,22)(10,18,15,21)(11,19,17,24)(12,20,16,23)(25,29)(26,34)(27,37)(28,31)(30,40)(32,39)(33,36)(35,38)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,d^2=gh,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8842080);