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 | |
4559 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)(9,10,14,16)(11,15,19,22)(12,20,21,17)(13,18,23,24)(25,26,31,35)(27,36,39,32)(28,33,40,30)(29,37,38,34);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,18)(13,17)(14,19)(16,22)(20,23)(21,24)(25,27)(26,32)(28,37)(29,33)(30,38)(31,39)(34,40)(35,36);(1,2,5,4)(3,6,8,7)(10,16)(11,19)(17,20)(18,24)(25,28,31,40)(26,30,35,33)(27,37,39,34)(29,32,38,36);(1,2,5,4)(3,6,8,7)(9,11)(10,15)(12,18)(13,17)(14,19)(16,22)(20,23)(21,24)(25,28)(26,33)(27,37)(29,32)(30,35)(31,40)(34,39)(36,38);(9,12,14,21)(10,17,16,20)(11,18,19,24)(13,22,23,15)(25,29,31,38)(26,34,35,37)(27,33,39,30)(28,32,40,36);(1,4,5,2)(3,7,8,6)(9,13)(10,18)(11,17)(12,15)(14,23)(16,24)(19,20)(21,22)(25,30,31,33)(26,28,35,40)(27,38,39,29)(32,37,36,34)Polycyclic presentation:
2^9|a^2=gi,b^2=ghi,c^2=g,d^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7784136);