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 | #59304 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4271 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,19)(12,22)(13,16)(14,20)(15,17)(18,23)(21,24)(25,26)(27,34)(28,32)(29,31)(30,33)(35,40)(36,39)(37,38);(1,2,5,4)(3,6,7,8)(9,11,15,13)(10,16,17,19)(12,20,23,21)(14,22,24,18)(27,28)(32,34)(35,36)(39,40);(2,4)(6,8)(10,17)(14,24)(16,19)(18,22)(25,27,31,28)(26,32,29,34)(30,36,38,35)(33,40,37,39);(3,7)(6,8)(9,12,15,23)(10,18,17,22)(11,21,13,20)(14,16,24,19)(25,28,31,27)(26,32,29,34)(30,36,38,35)(33,39,37,40);(1,3,5,7)(2,6,4,8)(9,13,15,11)(10,19,17,16)(12,21,23,20)(14,18,24,22)(25,29,31,26)(27,34,28,32)(30,37,38,33)(35,40,36,39);(1,4,5,2)(3,8,7,6)(9,14,15,24)(10,20,17,21)(11,22,13,18)(12,16,23,19)(25,30,31,38)(26,33,29,37)(27,35,28,36)(32,39,34,40)Polycyclic presentation:
2^9|b^2=hi,c^2=gh,d^2=h,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8249025);