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 | |
4441 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)(5,8)(6,7)(25,26,32,34)(27,29,31,28)(30,33,39,40)(35,37,38,36);(1,3)(2,5)(4,6)(7,8)(9,10)(11,17)(12,20)(13,14)(15,19)(16,21)(18,23)(22,24)(25,27,32,31)(26,28,34,29)(30,38,39,35)(33,37,40,36);(1,4)(2,8)(3,6)(5,7)(10,14)(12,21)(15,17)(18,22)(25,28,32,29)(26,27,34,31)(30,37,39,36)(33,38,40,35);(1,2,7,6)(3,4,8,5)(9,11)(10,15)(12,22)(13,19)(14,17)(16,24)(18,21)(20,23)(25,29,32,28)(26,27,34,31)(30,36,39,37)(33,38,40,35);(1,5,7,4)(2,8,6,3)(9,12)(10,16)(11,18)(13,21)(14,20)(15,23)(17,24)(19,22)(25,30,32,39)(26,33,34,40)(27,35,31,38)(28,36,29,37);(1,6,7,2)(3,4,8,5)(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,21)(22,24)(25,31,32,27)(26,28,34,29)(30,38,39,35)(33,36,40,37)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=gh,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8947608);