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 | |
4459 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)(3,4)(5,6)(7,8)(9,10,14,11)(12,15,19,17)(13,18,21,16)(20,23,24,22)(25,26,31,33)(27,32,36,29)(28,39,38,34)(30,40,37,35);(2,6)(3,7)(12,19)(15,17)(20,24)(22,23)(25,27)(26,32)(28,37)(29,33)(30,38)(31,36)(34,40)(35,39);(1,3,5,7)(2,4,6,8)(10,11)(15,17)(16,18)(22,23)(26,33)(27,36)(28,38)(35,40);(1,4)(2,7)(3,6)(5,8)(9,11,14,10)(12,17,19,15)(13,18,21,16)(20,23,24,22)(25,28,31,38)(26,34,33,39)(27,37,36,30)(29,35,32,40);(9,12,14,19)(10,15,11,17)(13,20,21,24)(16,22,18,23)(25,29)(26,27)(28,35)(30,39)(31,32)(33,36)(34,37)(38,40);(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,30,31,37)(26,35,33,40)(27,38,36,28)(29,34,32,39)Polycyclic presentation:
2^9|a^2=h,c^2=g,d^2=h,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9624933);