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 | |
4527 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,15)(12,20)(13,17)(14,19)(16,21)(18,22)(23,24)(25,26,31,30)(27,35,36,32)(28,37,39,33)(29,34,38,40);(3,7)(5,8)(9,10)(11,15)(12,16)(13,17)(14,19)(18,23)(20,21)(22,24)(25,27,31,36)(26,32,30,35)(28,38,39,29)(33,40,37,34);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,18)(13,19)(15,17)(16,22)(20,23)(21,24)(25,26,31,30)(27,32,36,35)(28,33,39,37)(29,34,38,40);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,18)(13,19)(14,17)(16,23)(20,22)(21,24)(25,28)(26,33)(27,29)(30,37)(31,39)(32,34)(35,40)(36,38);(1,2)(3,5)(4,6)(7,8)(9,12)(10,16)(11,18)(13,21)(14,22)(15,23)(17,20)(19,24)(25,29)(26,34)(27,28)(30,40)(31,38)(32,33)(35,37)(36,39);(9,10)(11,14)(12,20)(13,17)(15,19)(16,21)(18,23)(22,24)(25,30,31,26)(27,35,36,32)(28,33,39,37)(29,34,38,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8598128);