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 | |
4318 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,15,17)(11,19,13,16)(12,18,22,14)(20,21,23,24)(25,26)(27,36)(28,39)(29,30)(31,35)(32,38)(33,40)(34,37);(1,3,5,8)(2,6,4,7)(9,11,15,13)(10,16,17,19)(12,20,22,23)(14,21,18,24)(25,27,31,38)(26,32,35,36)(28,37,40,29)(30,33,34,39);(10,17)(12,22)(16,19)(20,23)(25,28,31,40)(26,33,35,39)(27,29,38,37)(30,36,34,32);(1,2,5,4)(3,6,8,7)(9,12)(10,18)(11,20)(13,23)(14,17)(15,22)(16,24)(19,21)(25,28)(26,33)(27,37)(29,38)(30,36)(31,40)(32,34)(35,39);(9,13,15,11)(10,19,17,16)(12,23,22,20)(14,24,18,21)(25,29,31,37)(26,34,35,30)(27,28,38,40)(32,39,36,33);(1,4,5,2)(3,7,8,6)(9,14,15,18)(10,12,17,22)(11,21,13,24)(16,20,19,23)(25,30,31,34)(26,29,35,37)(27,33,38,39)(28,36,40,32)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=i,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7805575);