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 | |
4195 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10,15,17)(11,16,21,22)(12,14,20,18)(13,23,24,19)(26,31)(27,34)(28,36)(37,39);(2,3)(6,7)(9,11)(10,16)(12,19)(13,14)(15,21)(17,22)(18,24)(20,23)(25,26,30,31)(27,29,34,32)(28,33,36,38)(35,37,40,39);(1,3,5,2)(4,7,8,6)(10,17)(11,21)(14,18)(19,23)(25,27,30,34)(26,32,31,29)(28,35,36,40)(33,39,38,37);(1,4)(2,6)(3,7)(5,8)(9,12,15,20)(10,18,17,14)(11,19,21,23)(13,16,24,22)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(9,13)(10,19)(11,18)(12,22)(14,21)(15,24)(16,20)(17,23)(25,28,30,36)(26,33,31,38)(27,35,34,40)(29,37,32,39);(9,14,15,18)(10,20,17,12)(11,13,21,24)(16,23,22,19)(25,29,30,32)(26,34,31,27)(28,37,36,39)(33,40,38,35)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=g,d^2=gi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8324511);