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 | |
4487 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,7,8,4)(9,10)(11,18)(12,20)(13,15)(14,17)(16,21)(19,23)(22,24)(25,26,30,35)(27,31,38,32)(28,33,34,29)(36,37,39,40);(1,3)(2,4)(5,8)(6,7)(9,11,14,13)(10,15,17,18)(12,19,21,22)(16,23,20,24)(25,27)(26,31)(28,39)(29,37)(30,38)(32,35)(33,40)(34,36);(1,2)(3,4)(5,6)(7,8)(25,27,30,38)(26,32,35,31)(28,36,34,39)(29,37,33,40);(2,6)(4,7)(9,12)(10,16)(11,19)(13,22)(14,21)(15,23)(17,20)(18,24)(25,27)(26,31)(28,39)(29,37)(30,38)(32,35)(33,40)(34,36);(1,4,5,7)(2,3,6,8)(9,13,14,11)(10,15,17,18)(12,19,21,22)(16,24,20,23)(25,28)(26,33)(27,36)(29,35)(30,34)(31,37)(32,40)(38,39);(1,3)(2,4)(5,8)(6,7)(9,10)(11,15)(12,16)(13,18)(14,17)(19,23)(20,21)(22,24)(25,29,30,33)(26,34,35,28)(27,37,38,40)(31,36,32,39)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=i,e^2=g,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10103246);