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 | |
4579 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,14)(15,18)(16,20)(22,24)(25,26)(27,31)(28,33)(29,36)(30,32)(34,40)(35,38)(37,39);(1,2)(3,4)(5,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,27,30,35)(26,31,32,38)(28,36,39,34)(29,37,40,33);(2,6)(3,7)(11,17)(15,18)(19,23)(22,24)(26,32)(27,35)(33,37)(34,36);(3,7)(4,8)(9,10)(11,18)(12,20)(13,14)(15,17)(16,21)(19,22)(23,24)(25,28)(26,33)(27,36)(29,31)(30,39)(32,37)(34,35)(38,40);(1,3)(2,4)(5,7)(6,8)(9,11,13,17)(10,15,14,18)(12,19,21,23)(16,22,20,24)(25,27)(26,31)(28,34)(29,37)(30,35)(32,38)(33,40)(36,39);(1,4)(2,3)(5,8)(6,7)(9,12,13,21)(10,16,14,20)(11,19,17,23)(15,22,18,24)(25,29,30,40)(26,34,32,36)(27,37,35,33)(28,38,39,31)Polycyclic presentation:
2^9|b^2=h,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8972510);