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 | |
4399 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)(4,7)(6,8)(9,10,15,19)(11,16,20,22)(12,14,18,24)(13,17,23,21)(25,26)(27,32)(28,33)(29,30)(31,35)(34,39)(36,37)(38,40);(1,2,6,8)(3,5,7,4)(9,11)(10,16)(12,21)(13,14)(15,20)(17,18)(19,22)(23,24)(25,27,31,33)(26,32,35,28)(29,36,34,40)(30,37,39,38);(1,3,6,7)(2,4,8,5)(11,20)(13,23)(16,22)(17,21)(25,28,31,32)(26,27,35,33)(29,38,34,37)(30,36,39,40);(1,4,6,5)(2,3,8,7)(9,12)(10,14)(11,21)(13,16)(15,18)(17,20)(19,24)(22,23)(25,29)(26,30)(27,36)(28,38)(31,34)(32,37)(33,40)(35,39);(1,2)(3,4)(5,7)(6,8)(9,13)(10,17)(11,14)(12,22)(15,23)(16,18)(19,21)(20,24)(25,28)(26,33)(27,35)(29,37)(30,36)(31,32)(34,38)(39,40);(1,5,6,4)(2,7,8,3)(9,14)(10,18)(11,13)(12,19)(15,24)(16,17)(20,23)(21,22)(25,30)(26,34)(27,37)(28,36)(29,35)(31,39)(32,40)(33,38)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=h,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8973274);