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 | |
4531 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,13,17)(11,15,19,14)(12,16,20,23)(18,22,24,21)(25,26)(27,28)(29,37)(30,31)(32,40)(33,36)(34,39)(35,38);(1,2,5,6)(3,7,8,4)(9,11,13,19)(10,14,17,15)(12,18,20,24)(16,21,23,22)(25,27,30,33)(26,28,31,36)(29,38,40,34)(32,39,37,35);(2,6)(4,7)(12,20)(16,23)(18,24)(21,22)(26,31)(27,33)(32,37)(34,38);(1,3,5,8)(2,7,6,4)(9,10)(11,14)(12,16)(13,17)(15,19)(18,21)(20,23)(22,24)(25,28)(26,27)(29,39)(30,36)(31,33)(32,38)(34,37)(35,40);(1,3)(2,4)(5,8)(6,7)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,29,30,40)(26,32,31,37)(27,34,33,38)(28,35,36,39);(1,4)(2,3)(5,7)(6,8)(9,12,13,20)(10,16,17,23)(11,18,19,24)(14,21,15,22)(25,26)(27,28)(29,32)(30,31)(33,36)(34,35)(37,40)(38,39)Polycyclic presentation:
2^9|a^2=g,b^2=gi,d^2=hi,e^2=h,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=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8265956);