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 | |
4536 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10,14,13)(11,17,19,15)(12,16,22,21)(18,24,20,23);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,18)(13,17)(14,19)(16,23)(20,22)(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)(10,13)(15,17)(16,21)(23,24)(26,31)(29,38)(32,39)(36,40);(9,11)(10,15)(12,20)(13,17)(14,19)(16,24)(18,22)(21,23)(25,27,30,33)(26,29,31,38)(28,35,37,34)(32,40,39,36);(1,4,5,3)(2,8,6,7)(9,12)(10,16)(11,18)(13,21)(14,22)(15,23)(17,24)(19,20)(25,28,30,37)(26,32,31,39)(27,34,33,35)(29,36,38,40);(9,13,14,10)(11,17,19,15)(12,21,22,16)(18,24,20,23)(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=i,c^2=gh,d^2=i,e^2=ghi,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, 7559977);