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 | |
4250 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)(11,12,17,20)(13,16,21,22)(18,19,23,24)(25,26)(27,36)(28,39)(29,40)(30,35)(31,32)(33,38)(34,37);(1,3,5,8)(2,6,4,7)(25,27)(26,30)(28,37)(29,38)(31,35)(32,36)(33,34)(39,40);(1,2,5,4)(3,6,8,7)(10,15)(11,17)(16,22)(18,23)(26,32)(27,35)(28,38)(34,40);(9,11)(10,12)(13,18)(14,17)(15,20)(16,19)(21,23)(22,24)(25,28,31,38)(26,33,32,39)(27,37,35,29)(30,34,36,40);(9,12)(10,11)(13,19)(14,20)(15,17)(16,18)(21,24)(22,23)(25,29,31,37)(26,34,32,40)(27,38,35,28)(30,33,36,39);(1,4,5,2)(3,7,8,6)(9,13,14,21)(10,16,15,22)(11,18,17,23)(12,19,20,24)(25,30,31,36)(26,35,32,27)(28,34,38,40)(29,39,37,33)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=gi,d^2=i,e^2=i,f^2=ghi,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, 7854275);