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 |
| prime radical | #51775 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4315 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(25,26,30,31)(27,33,36,32)(28,34,37,40)(29,38,39,35);(1,3,5,8)(2,6,4,7)(9,10,14,15)(11,16,20,23)(12,13,17,22)(18,24,21,19)(26,31)(28,37)(32,33)(35,38);(10,15)(13,22)(16,23)(19,24)(25,27,30,36)(26,32,31,33)(28,35,37,38)(29,34,39,40);(1,2,5,4)(3,6,8,7)(9,11)(10,16)(12,21)(13,19)(14,20)(15,23)(17,18)(22,24)(25,27)(26,33)(28,35)(29,34)(30,36)(31,32)(37,38)(39,40);(1,2,5,4)(3,6,8,7)(9,12,14,17)(10,13,15,22)(11,18,20,21)(16,24,23,19)(25,28)(26,34)(27,35)(29,33)(30,37)(31,40)(32,39)(36,38);(1,4,5,2)(3,7,8,6)(9,13)(10,17)(11,19)(12,15)(14,22)(16,18)(20,24)(21,23)(25,29)(26,35)(27,34)(28,33)(30,39)(31,38)(32,37)(36,40)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=h,d^2=gh,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fi,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7919207);