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 | |
4493 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10,14,18)(11,13,17,15)(12,20,22,16)(19,23,24,21)(25,26)(27,36)(28,33)(29,34)(30,32)(31,35)(37,38)(39,40);(1,2)(3,6)(4,7)(5,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,23)(20,21)(22,24)(25,27,31,30)(26,32,35,36)(28,37,34,39)(29,38,33,40);(1,3,5,4)(2,6,8,7)(9,11)(10,15)(12,19)(13,18)(14,17)(16,23)(20,21)(22,24)(25,26)(27,36)(28,33)(29,34)(30,32)(31,35)(37,38)(39,40);(1,4,5,3)(2,7,8,6)(9,12,14,22)(10,16,18,20)(11,19,17,24)(13,21,15,23)(25,28,31,34)(26,33,35,29)(27,37,30,39)(32,40,36,38);(9,11)(10,13)(12,19)(14,17)(15,18)(16,21)(20,23)(22,24)(25,29,31,33)(26,34,35,28)(27,38,30,40)(32,39,36,37);(9,13)(10,17)(11,18)(12,21)(14,15)(16,24)(19,20)(22,23)(25,30,31,27)(26,32,35,36)(28,39,34,37)(29,38,33,40)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=ghi,d^2=g,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^e=fhi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7562540);