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 | |
4346 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(10,15)(11,17)(13,21)(19,24)(26,30)(28,35)(31,36)(34,39);(2,6)(3,7)(9,10,14,15)(11,12,17,20)(13,22,21,16)(18,24,23,19)(27,33)(31,36)(34,39)(38,40);(1,3)(2,4)(5,7)(6,8)(9,11,14,17)(10,12,15,20)(13,18,21,23)(16,19,22,24)(25,26,29,30)(27,31,33,36)(28,32,35,37)(34,38,39,40);(1,4)(2,3)(5,8)(6,7)(9,11)(10,12)(13,18)(14,17)(15,20)(16,19)(21,23)(22,24);(9,12)(10,11)(13,19)(14,20)(15,17)(16,18)(21,24)(22,23)(25,27)(26,31)(28,34)(29,33)(30,36)(32,38)(35,39)(37,40);(9,13,14,21)(10,16,15,22)(11,18,17,23)(12,19,20,24)(25,28)(26,32)(27,34)(29,35)(30,37)(31,38)(33,39)(36,40)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8717497);