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 | |
4414 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)(11,19)(12,21)(13,18)(14,16)(15,22)(17,23)(20,24)(25,26,30,32)(27,31,35,40)(28,33,37,38)(29,36,39,34);(1,2)(3,7)(4,5)(6,8)(9,11,14,22)(10,15,16,19)(12,18,23,20)(13,21,24,17)(25,27)(26,31)(28,36)(29,38)(30,35)(32,40)(33,39)(34,37);(2,5)(6,7)(10,16)(13,24)(15,19)(17,21)(26,32)(27,35)(33,38)(34,36);(9,12,14,23)(10,17,16,21)(11,20,22,18)(13,15,24,19)(25,28)(26,33)(27,34)(29,31)(30,37)(32,38)(35,36)(39,40);(1,3,4,8)(2,6,5,7)(9,11)(10,15)(12,20)(13,21)(14,22)(16,19)(17,24)(18,23)(25,29)(26,34)(27,33)(28,31)(30,39)(32,36)(35,38)(37,40);(1,2)(3,6)(4,5)(7,8)(9,13,14,24)(10,18,16,20)(11,21,22,17)(12,15,23,19)(25,27)(26,31)(28,34)(29,33)(30,35)(32,40)(36,37)(38,39)Polycyclic presentation:
2^9|a^2=gh,b^2=g,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, 8121708);