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 | |
4648 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(9,10,14,17)(11,16,20,15)(12,21,22,13)(18,19,23,24);(3,7)(6,8)(9,11)(10,15)(12,18)(13,19)(14,20)(16,17)(21,24)(22,23)(25,26,30,31)(27,32,34,39)(28,29,33,38)(35,40,37,36);(1,2,5,4)(3,6,7,8)(12,22)(13,21)(18,23)(19,24)(26,31)(27,34)(29,38)(35,37);(9,12)(10,13)(11,18)(14,22)(15,19)(16,24)(17,21)(20,23)(25,27)(26,32)(28,37)(29,36)(30,34)(31,39)(33,35)(38,40);(1,3,5,7)(2,6,4,8)(9,11)(10,16)(12,23)(13,19)(14,20)(15,17)(18,22)(21,24)(25,28,30,33)(26,29,31,38)(27,35,34,37)(32,40,39,36);(1,4,5,2)(3,8,7,6)(9,13)(10,12)(11,19)(14,21)(15,18)(16,23)(17,22)(20,24)(25,29)(26,33)(27,36)(28,31)(30,38)(32,35)(34,40)(37,39)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=h,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8826428);