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 | |
4368 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)(27,32,36,39)(28,38,35,30)(29,34,40,37);(2,6)(3,7)(12,19)(15,17)(20,24)(22,23)(25,27)(26,32)(28,29)(30,37)(31,36)(33,39)(34,38)(35,40);(1,3,5,7)(2,4,6,8)(10,11)(15,17)(16,18)(22,23)(26,33)(27,36)(28,35)(34,37);(1,4)(2,7)(3,6)(5,8)(9,11,14,10)(12,17,19,15)(13,18,21,16)(20,23,24,22)(25,28,31,35)(26,30,33,38)(27,29,36,40)(32,37,39,34);(9,12,14,19)(10,15,11,17)(13,20,21,24)(16,22,18,23)(25,29)(26,34)(27,35)(28,36)(30,32)(31,40)(33,37)(38,39);(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,30)(26,35)(27,37)(28,33)(29,39)(31,38)(32,40)(34,36)Polycyclic presentation:
2^9|a^2=h,c^2=g,d^2=h,e^2=ghi,f^2=ghi,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, 9624935);