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 | |
4516 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)(3,5)(4,6)(7,8)(9,10,15,18)(11,16,22,24)(12,17,20,21)(13,23,19,14)(25,26,31,35)(27,29,36,32)(28,37,39,33)(30,40,38,34);(2,3)(6,7)(9,11)(10,16)(12,13)(14,21)(15,22)(17,23)(18,24)(19,20)(25,27,31,36)(26,32,35,29)(28,34,39,40)(30,37,38,33);(9,12)(10,17)(11,19)(13,22)(14,16)(15,20)(18,21)(23,24)(25,28,31,39)(26,33,35,37)(27,34,36,40)(29,38,32,30);(2,3)(4,8)(9,12)(10,17)(11,13)(14,24)(15,20)(16,23)(18,21)(19,22)(25,29)(26,27)(28,38)(30,39)(31,32)(33,34)(35,36)(37,40);(1,3,5,2)(4,7,8,6)(9,13)(10,14)(11,20)(12,22)(15,19)(16,17)(18,23)(21,24)(25,30,31,38)(26,34,35,40)(27,33,36,37)(28,29,39,32);(1,4)(2,6)(3,7)(5,8)(9,14)(10,13)(11,21)(12,16)(15,23)(17,22)(18,19)(20,24)(25,26)(27,32)(28,33)(29,36)(30,34)(31,35)(37,39)(38,40)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10319489);