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 | |
4379 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)(9,10,14,15)(11,13,17,22)(12,16,21,23)(18,20,19,24)(25,26,31,36)(27,37,38,32)(28,30,35,33)(29,34,40,39);(1,3,5,8)(2,6,4,7)(25,27,31,38)(26,32,36,37)(28,29,35,40)(30,39,33,34);(1,4,5,2)(3,7,8,6)(10,15)(13,22)(16,23)(20,24)(27,38)(29,40)(32,37)(34,39);(9,11,14,17)(10,13,15,22)(12,19,21,18)(16,24,23,20)(25,28)(26,33)(27,29)(30,36)(31,35)(32,34)(37,39)(38,40);(9,12,14,21)(10,16,15,23)(11,18,17,19)(13,20,22,24)(25,29)(26,34)(27,35)(28,38)(30,32)(31,40)(33,37)(36,39);(9,13)(10,17)(11,15)(12,20)(14,22)(16,19)(18,23)(21,24)(25,30,31,33)(26,35,36,28)(27,39,38,34)(29,32,40,37)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=gi,d^2=h,e^2=h,f^2=i,b^a=bg,c^a=ch,d^a=di,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7976688);