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 |
| prime radical | #37729 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4264 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(10,16)(11,12)(18,21)(19,20)(25,26)(27,35)(28,29)(30,32)(31,34)(33,40)(36,37)(38,39);(1,3,4,8)(2,5,6,7)(9,10)(11,17)(12,14)(13,21)(15,16)(18,23)(19,22)(20,24)(25,27,31,30)(26,32,34,35)(28,38,33,36)(29,37,40,39);(9,11,15,12)(10,17,16,14)(13,19,23,20)(18,24,21,22)(25,28,31,33)(26,29,34,40)(27,36,30,38)(32,39,35,37);(1,2,4,6)(3,5,8,7)(9,12,15,11)(10,17,16,14)(13,20,23,19)(18,24,21,22)(25,29)(26,33)(27,37)(28,34)(30,39)(31,40)(32,36)(35,38);(1,3)(2,5)(4,8)(6,7)(9,13,15,23)(10,18,16,21)(11,19,12,20)(14,22,17,24);(1,2)(3,5)(4,6)(7,8)(9,14,15,17)(10,12,16,11)(13,22,23,24)(18,20,21,19)(25,30,31,27)(26,32,34,35)(28,38,33,36)(29,39,40,37)Polycyclic presentation:
2^9|b^2=gh,c^2=hi,d^2=gi,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8411992);