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 | |
4540 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10,15,14)(11,17,12,16)(13,18,22,21)(19,24,20,23);(1,2)(3,7)(4,8)(5,6)(9,11,15,12)(10,16,14,17)(13,19,22,20)(18,23,21,24)(25,26,30,31)(27,29,33,38)(28,32,37,39)(34,36,35,40);(1,3,5,4)(2,7,6,8)(10,14)(16,17)(18,21)(23,24)(26,31)(29,38)(32,39)(36,40);(1,4,5,3)(2,8,6,7)(9,12,15,11)(10,17,14,16)(13,19,22,20)(18,23,21,24)(25,27,30,33)(26,29,31,38)(28,35,37,34)(32,40,39,36);(9,13)(10,18)(11,19)(12,20)(14,21)(15,22)(16,23)(17,24)(25,28,30,37)(26,32,31,39)(27,34,33,35)(29,36,38,40);(9,14,15,10)(11,17,12,16)(13,21,22,18)(19,24,20,23)(25,29)(26,33)(27,31)(28,36)(30,38)(32,35)(34,39)(37,40)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=gh,d^2=gi,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,e^d=ehi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7559942);