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 | |
4555 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,13,14)(11,15,17,18)(12,16,20,21)(19,22,23,24)(25,26,30,31)(27,28,33,32)(29,37,38,34)(35,39,40,36);(1,2)(3,7)(4,5)(6,8)(10,14)(12,20)(15,18)(19,23)(26,31)(27,33)(29,38)(36,39);(2,5)(6,7)(11,17)(15,18)(19,23)(22,24)(26,31)(28,32)(34,37)(36,39);(2,5)(6,7)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,27,30,33)(26,32,31,28)(29,35,38,40)(34,39,37,36);(1,2)(3,6)(4,5)(7,8)(9,11,13,17)(10,15,14,18)(12,19,20,23)(16,22,21,24)(25,28,30,32)(26,33,31,27)(29,36,38,39)(34,40,37,35);(1,3)(2,6)(4,8)(5,7)(9,12,13,20)(10,16,14,21)(11,19,17,23)(15,22,18,24)(25,29)(26,34)(27,35)(28,36)(30,38)(31,37)(32,39)(33,40)Polycyclic presentation:
2^9|a^2=g,d^2=h,e^2=g,f^2=gh,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, 10230206);