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 | |
4455 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)(3,5)(4,7)(6,8)(9,10,14,19)(11,17,22,15)(12,13,18,16)(20,24,23,21);(1,2,6,8)(3,5,7,4)(9,11)(10,15)(12,20)(13,21)(14,22)(16,24)(17,19)(18,23)(25,26,30,31)(27,28,32,36)(29,33,37,38)(34,35,39,40);(1,3,6,7)(2,4,8,5)(12,18)(13,16)(20,23)(21,24)(26,31)(27,32)(29,37)(35,40);(1,4,6,5)(2,3,8,7)(9,12,14,18)(10,16,19,13)(11,20,22,23)(15,24,17,21)(25,27)(26,28)(29,34)(30,32)(31,36)(33,35)(37,39)(38,40);(1,2)(3,4)(5,7)(6,8)(9,11)(10,17)(12,23)(13,21)(14,22)(15,19)(16,24)(18,20)(25,28)(26,32)(27,31)(29,35)(30,36)(33,39)(34,38)(37,40);(1,5,6,4)(2,7,8,3)(9,13,14,16)(10,18,19,12)(11,21,22,24)(15,23,17,20)(25,29)(26,33)(27,34)(28,35)(30,37)(31,38)(32,39)(36,40)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=h,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8815609);