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 |
| prime radical | #42999 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4548 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)(3,4)(5,6)(7,8)(9,10,15,17)(11,16,20,24)(12,14,19,22)(13,21,23,18)(25,26,30,31)(27,32,35,38)(28,36,37,29)(33,39,40,34);(2,6)(3,7)(9,11,15,20)(10,16,17,24)(12,18,19,21)(13,22,23,14)(26,31)(29,36)(32,38)(34,39);(1,3,5,7)(2,4,6,8)(10,17)(11,20)(14,22)(18,21)(25,27,30,35)(26,32,31,38)(28,33,37,40)(29,34,36,39);(1,4)(2,7)(3,6)(5,8)(9,12,15,19)(10,14,17,22)(11,21,20,18)(13,16,23,24)(25,28)(26,29)(27,33)(30,37)(31,36)(32,34)(35,40)(38,39);(9,13)(10,18)(11,14)(12,16)(15,23)(17,21)(19,24)(20,22)(25,29)(26,28)(27,34)(30,36)(31,37)(32,33)(35,39)(38,40);(9,14)(10,19)(11,13)(12,17)(15,22)(16,21)(18,24)(20,23)(25,27)(26,32)(28,33)(29,34)(30,35)(31,38)(36,39)(37,40)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=g,d^2=h,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, 8109366);