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 | |
4421 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,6,8,4)(9,10,13,16)(11,17,19,14)(12,15,21,24)(18,23,20,22)(25,26,30,35)(27,31,38,32)(28,33,34,29)(36,37,39,40);(1,3)(2,4)(5,8)(6,7)(9,11)(10,14)(12,20)(13,19)(15,23)(16,17)(18,21)(22,24)(25,27)(26,31)(28,39)(29,37)(30,38)(32,35)(33,40)(34,36);(1,2)(3,4)(5,7)(6,8)(9,11,13,19)(10,14,16,17)(12,18,21,20)(15,22,24,23)(25,27,30,38)(26,32,35,31)(28,36,34,39)(29,37,33,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,21)(23,24)(25,27)(26,32)(28,36)(29,37)(30,38)(31,35)(33,40)(34,39);(1,4,5,6)(2,3,7,8)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,28)(26,33)(27,36)(29,35)(30,34)(31,37)(32,40)(38,39);(1,3)(2,4)(5,8)(6,7)(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,23)(21,24)(25,29,30,33)(26,34,35,28)(27,37,38,40)(31,36,32,39)Polycyclic presentation:
2^9|a^2=h,c^2=i,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9627407);