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 | |
4419 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,13,17,22)(12,16,20,23)(18,21,24,19)(25,26,31,34)(27,35,37,32)(28,38,39,30)(29,33,40,36);(2,3)(6,7)(9,11)(10,13)(12,18)(14,17)(15,22)(16,21)(19,23)(20,24)(25,27,31,37)(26,32,34,35)(28,36,39,33)(29,38,40,30);(1,2,5,3)(4,6,8,7)(10,15)(11,17)(16,23)(18,24);(9,11)(10,13)(12,18)(14,17)(15,22)(16,21)(19,23)(20,24)(25,28)(26,30)(27,36)(29,32)(31,39)(33,37)(34,38)(35,40);(1,3,5,2)(4,6,8,7)(9,12)(10,16)(11,18)(13,21)(14,20)(15,23)(17,24)(19,22)(25,29,31,40)(26,33,34,36)(27,30,37,38)(28,32,39,35);(1,4)(2,6)(3,7)(5,8)(9,13)(10,11)(12,19)(14,22)(15,17)(16,24)(18,23)(20,21)(25,30)(26,28)(27,29)(31,38)(32,36)(33,35)(34,39)(37,40)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=hi,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9755806);