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 | #35190 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4322 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10,14,15)(11,19,20,16)(12,22,23,17)(13,21,24,18)(25,26,31,29)(27,28,34,32)(30,37,38,33)(35,39,40,36);(1,3,5,8)(2,6,7,4)(10,15)(13,24)(16,19)(17,22)(26,29)(27,34)(33,37)(35,40);(1,3)(2,4)(5,8)(6,7)(10,15)(11,20)(17,22)(18,21)(25,27,31,34)(26,32,29,28)(30,35,38,40)(33,39,37,36);(1,4)(2,3)(5,6)(7,8)(9,11,14,20)(10,16,15,19)(12,21,23,18)(13,17,24,22)(25,28)(26,27)(29,34)(30,36)(31,32)(33,35)(37,40)(38,39);(9,12)(10,17)(11,21)(13,19)(14,23)(15,22)(16,24)(18,20)(25,29,31,26)(27,28,34,32)(30,37,38,33)(35,36,40,39);(9,13,14,24)(10,18,15,21)(11,17,20,22)(12,19,23,16)(25,30,31,38)(26,33,29,37)(27,35,34,40)(28,36,32,39)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=ghi,d^2=gi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8664833);