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 | |
4504 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(4,8)(6,7)(9,10,14,15)(11,18,16,12)(13,21,22,17)(19,20,23,24)(26,32)(28,36)(30,34)(35,37);(1,2,5,3)(4,6,8,7)(10,15)(11,16)(17,21)(19,23)(25,26)(27,30)(28,33)(29,34)(31,32)(35,38)(36,39)(37,40);(2,3)(6,7)(10,15)(11,16)(17,21)(19,23)(25,27,31,29)(26,30,32,34)(28,37,36,35)(33,40,39,38);(1,3,5,2)(4,7,8,6)(9,11,14,16)(10,12,15,18)(13,19,22,23)(17,20,21,24)(25,28,31,36)(26,33,32,39)(27,35,29,37)(30,38,34,40);(9,12)(10,16)(11,15)(13,20)(14,18)(17,23)(19,21)(22,24)(25,29,31,27)(26,34,32,30)(28,37,36,35)(33,40,39,38);(1,4)(2,6)(3,7)(5,8)(9,13,14,22)(10,17,15,21)(11,19,16,23)(12,20,18,24)(25,30,31,34)(26,27,32,29)(28,38,36,40)(33,35,39,37)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=gh,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8014693);