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×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4290 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)(11,12)(13,21)(14,18)(15,16)(17,23)(19,24)(20,22)(25,26)(27,30)(28,32)(29,31)(33,35)(34,38)(36,37)(39,40);(1,2,5,6)(3,7,4,8)(9,11,15,14)(10,12,16,18)(13,19,23,22)(17,20,21,24)(25,27)(26,30)(28,34)(29,35)(31,33)(32,38)(36,39)(37,40);(1,3,5,4)(2,7,6,8)(10,16)(12,18)(17,21)(20,24)(26,29)(27,33)(28,36)(38,40);(9,12,15,18)(10,11,16,14)(13,20,23,24)(17,19,21,22)(25,28,31,36)(26,32,29,37)(27,34,33,39)(30,38,35,40);(1,3,5,4)(2,7,6,8)(9,13,15,23)(10,17,16,21)(11,19,14,22)(12,20,18,24)(25,29,31,26)(27,35,33,30)(28,37,36,32)(34,40,39,38);(1,4,5,3)(2,8,6,7)(9,14,15,11)(10,18,16,12)(13,22,23,19)(17,24,21,20)(25,30,31,35)(26,33,29,27)(28,38,36,40)(32,39,37,34)Polycyclic presentation:
2^9|b^2=ghi,c^2=g,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7617089);