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 | |
4404 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(25,26)(27,32)(28,33)(29,31)(30,34)(35,38)(36,40)(37,39);(2,3)(6,7)(9,10)(11,16)(12,13)(14,15)(17,21)(18,22)(19,24)(20,23)(25,27,31,34)(26,32,29,30)(28,36,39,35)(33,40,37,38);(1,3,5,2)(4,7,8,6)(10,15)(12,21)(16,23)(18,24)(26,29)(27,34)(33,37)(35,36);(1,4)(2,6)(3,7)(5,8)(9,11)(10,16)(12,18)(13,22)(14,20)(15,23)(17,19)(21,24)(25,28,31,39)(26,33,29,37)(27,35,34,36)(30,40,32,38);(9,12)(10,17)(11,18)(13,15)(14,21)(16,19)(20,24)(22,23)(25,29,31,26)(27,32,34,30)(28,37,39,33)(35,38,36,40);(9,13,14,17)(10,12,15,21)(11,19,20,22)(16,24,23,18)(25,30,31,32)(26,34,29,27)(28,38,39,40)(33,35,37,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8540497);