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 | #44086 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4291 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,16)(12,18)(13,19)(14,20)(15,17)(21,24)(22,23)(25,26,31,32)(27,35,39,33)(28,34,30,29)(36,40,38,37);(1,2,5,6)(3,4,7,8)(9,11)(10,16)(12,13)(14,17)(15,20)(18,19)(21,22)(23,24)(26,32)(28,30)(33,35)(36,38);(2,6)(3,7)(10,17)(11,20)(13,21)(18,23)(25,27,31,39)(26,33,32,35)(28,36,30,38)(29,40,34,37);(2,6)(3,7)(9,12)(10,18)(11,21)(13,20)(14,19)(15,22)(16,24)(17,23)(25,28,31,30)(26,29,32,34)(27,36,39,38)(33,40,35,37);(1,3)(2,4)(5,7)(6,8)(9,13,15,21)(10,19,17,24)(11,12,20,22)(14,23,16,18)(25,29,31,34)(26,28,32,30)(27,37,39,40)(33,38,35,36);(1,4)(2,3)(5,8)(6,7)(9,14)(10,11)(12,19)(13,23)(15,16)(17,20)(18,21)(22,24)(25,30,31,28)(26,34,32,29)(27,38,39,36)(33,37,35,40)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=g,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10236598);