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 | #36961 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4362 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10,15,18)(11,17,12,16)(13,14,23,24)(19,22,21,20)(25,26,29,32)(27,30,33,37)(28,35,36,31)(34,40,39,38);(1,3,5,8)(2,6,7,4)(9,11,15,12)(10,16,18,17)(13,19,23,21)(14,20,24,22)(27,33)(30,37)(34,39)(38,40);(1,3)(2,4)(5,8)(6,7)(9,11,15,12)(10,17,18,16)(13,21,23,19)(14,20,24,22)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,36)(39,40);(1,4)(2,3)(5,6)(7,8)(9,12,15,11)(10,17,18,16)(13,21,23,19)(14,22,24,20)(25,27,29,33)(26,30,32,37)(28,34,36,39)(31,38,35,40);(9,13)(10,14)(11,19)(12,21)(15,23)(16,20)(17,22)(18,24)(25,28,29,36)(26,31,32,35)(27,34,33,39)(30,38,37,40);(9,14)(10,13)(11,20)(12,22)(15,24)(16,19)(17,21)(18,23)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=gi,d^2=ghi,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8026422);