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 | |
4558 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,6,8,7)(9,10,15,18)(11,16,19,14)(12,13,17,24)(20,21,22,23)(25,26,32,36)(27,37,39,31)(28,35,30,33)(29,40,38,34);(1,2)(3,5)(4,6)(7,8)(9,11,15,19)(10,16,18,14)(12,22,17,20)(13,23,24,21)(25,27,32,39)(26,31,36,37)(28,38,30,29)(33,34,35,40);(4,8)(6,7)(11,19)(14,16)(20,22)(21,23)(25,28,32,30)(26,33,36,35)(27,38,39,29)(31,34,37,40);(2,3)(6,7)(9,12)(10,13)(11,20)(14,23)(15,17)(16,21)(18,24)(19,22)(25,29)(26,34)(27,28)(30,39)(31,35)(32,38)(33,37)(36,40);(1,3,5,2)(4,7,8,6)(9,13)(10,17)(11,21)(12,18)(14,20)(15,24)(16,22)(19,23)(25,30,32,28)(26,35,36,33)(27,38,39,29)(31,34,37,40);(1,4,5,8)(2,6,3,7)(9,14)(10,11)(12,23)(13,20)(15,16)(17,21)(18,19)(22,24)(25,31,32,37)(26,27,36,39)(28,40,30,34)(29,35,38,33)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,c^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8296476);