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 | |
4311 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,16)(11,12)(14,24)(18,21)(25,26,30,34)(27,28,32,31)(29,33,38,40)(35,36,37,39);(1,2)(3,7)(4,5)(6,8)(9,10,15,16)(11,17,12,20)(13,14,23,24)(18,22,21,19)(25,27,30,32)(26,31,34,28)(29,37,38,35)(33,36,40,39);(2,5)(6,7)(9,11,15,12)(10,17,16,20)(13,21,23,18)(14,19,24,22)(25,26,30,34)(27,31,32,28)(29,33,38,40)(35,39,37,36);(2,5)(6,7)(9,12,15,11)(10,17,16,20)(13,21,23,18)(14,22,24,19)(25,27)(26,31)(28,34)(29,35)(30,32)(33,39)(36,40)(37,38);(1,2)(3,6)(4,5)(7,8)(9,13)(10,14)(11,18)(12,21)(15,23)(16,24)(17,22)(19,20)(25,28,30,31)(26,32,34,27)(29,36,38,39)(33,37,40,35);(1,3)(2,6)(4,8)(5,7)(9,14)(10,13)(11,19)(12,22)(15,24)(16,23)(17,21)(18,20)(25,29,30,38)(26,33,34,40)(27,35,32,37)(28,36,31,39)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=g,d^2=h,e^2=gh,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, 10231284);