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 | C2×#532 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4266 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10,15,19)(11,13,17,24)(12,18,14,16)(20,23,21,22)(25,26,29,32)(27,33,34,30)(28,31,36,38)(35,39,40,37);(1,2,5,6)(3,7,4,8)(9,11,15,17)(10,13,19,24)(12,20,14,21)(16,22,18,23)(27,34)(30,33)(35,40)(37,39);(9,12,15,14)(10,16,19,18)(11,20,17,21)(13,22,24,23)(25,27)(26,30)(28,35)(29,34)(31,37)(32,33)(36,40)(38,39);(1,3,5,4)(2,7,6,8)(9,10)(11,13)(12,16)(14,18)(15,19)(17,24)(20,22)(21,23)(25,26)(27,30)(28,31)(29,32)(33,34)(35,37)(36,38)(39,40);(1,4,5,3)(2,8,6,7)(9,13)(10,17)(11,19)(12,22)(14,23)(15,24)(16,21)(18,20);(9,14,15,12)(10,18,19,16)(11,21,17,20)(13,23,24,22)(25,28,29,36)(26,31,32,38)(27,35,34,40)(30,37,33,39)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=hi,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7553127);