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 | |
4501 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,13,18,16)(12,20,21,17)(19,23,24,22)(25,26)(27,33)(28,38)(29,34)(30,35)(31,32)(36,37)(39,40);(1,2)(3,7)(4,5)(6,8)(10,15)(11,18)(12,21)(22,23)(26,32)(29,35)(33,38)(36,39);(2,5)(6,7)(10,15)(13,16)(17,20)(22,23)(25,27,31,28)(26,33,32,38)(29,36,35,39)(30,40,34,37);(9,11,14,18)(10,16,15,13)(12,19,21,24)(17,23,20,22)(25,28,31,27)(26,33,32,38)(29,39,35,36)(30,40,34,37);(1,3,4,8)(2,6,5,7)(9,12)(10,17)(11,19)(13,22)(14,21)(15,20)(16,23)(18,24)(25,29,31,35)(26,34,32,30)(27,36,28,39)(33,37,38,40);(1,2)(3,6)(4,5)(7,8)(9,13,14,16)(10,18,15,11)(12,22,21,23)(17,24,20,19)(25,30,31,34)(26,35,32,29)(27,37,28,40)(33,36,38,39)Polycyclic presentation:
2^9|a^2=h,c^2=gh,d^2=g,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8121837);