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) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,14,17)(11,15,18,22)(12,13,16,21)(19,20,23,24)(25,26)(27,34)(28,37)(29,35)(30,32)(31,36)(33,38)(39,40);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,19)(13,20)(14,18)(16,23)(17,22)(21,24)(25,27)(26,31)(28,35)(29,33)(30,36)(32,34)(37,39)(38,40);(1,2,5,4)(3,6,8,7)(11,18)(15,22)(19,23)(20,24)(26,32)(28,38)(31,34)(35,40);(1,2,5,4)(3,6,8,7)(9,11)(10,15)(12,19)(13,20)(14,18)(16,23)(17,22)(21,24)(25,28,30,38)(26,33,32,37)(27,35,36,40)(29,34,39,31);(1,4,5,2)(3,7,8,6)(9,12)(10,13)(11,19)(14,16)(15,20)(17,21)(18,23)(22,24)(25,27)(26,34)(28,35)(29,37)(30,36)(31,32)(33,39)(38,40);(9,13)(10,16)(11,20)(12,17)(14,21)(15,23)(18,24)(19,22)(25,29,30,39)(26,35,32,40)(27,33,36,37)(28,34,38,31)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=g,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^c=dhi,f^e=fhi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7569240);