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 | #38668 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4256 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(10,11)(13,21)(15,17)(19,24)(25,26,30,31)(27,29,34,32)(28,36,38,33)(35,39,40,37);(3,7)(6,8)(9,10,14,11)(12,15,20,17)(13,16,21,18)(19,22,24,23)(26,31)(27,34)(33,36)(35,40);(1,2)(3,6)(4,5)(7,8)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(1,3)(2,6)(4,7)(5,8)(9,11,14,10)(12,15,20,17)(13,18,21,16)(19,22,24,23)(25,27,30,34)(26,32,31,29)(28,35,38,40)(33,39,36,37);(1,2)(3,6)(4,5)(7,8)(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,28,30,38)(26,33,31,36)(27,35,34,40)(29,37,32,39);(9,13)(10,16)(11,18)(12,19)(14,21)(15,22)(17,23)(20,24)(25,29,30,32)(26,34,31,27)(28,37,38,39)(33,40,36,35)Polycyclic presentation:
2^9|a^2=i,b^2=gi,d^2=g,e^2=g,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8059370);