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 | #39961 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4372 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,4)(6,7)(9,10,15,17)(11,16,14,21)(12,18,22,23)(13,19,24,20)(25,26,31,36)(27,33,28,32)(29,34,39,37)(30,38,40,35);(1,2,5,8)(3,6,4,7)(9,11,15,14)(10,16,17,21)(12,20,22,19)(13,23,24,18)(25,27,31,28)(26,32,36,33)(29,38,39,35)(30,37,40,34);(1,3,5,4)(2,6,8,7)(10,17)(11,14)(18,23)(19,20)(25,27,31,28)(26,32,36,33)(29,35,39,38)(30,34,40,37);(9,12)(10,18)(11,20)(13,16)(14,19)(15,22)(17,23)(21,24)(25,28,31,27)(26,33,36,32)(29,35,39,38)(30,34,40,37);(1,2)(3,6)(4,7)(5,8)(9,13)(10,19)(11,18)(12,21)(14,23)(15,24)(16,22)(17,20)(25,29,31,39)(26,34,36,37)(27,35,28,38)(30,32,40,33);(1,4,5,3)(2,7,8,6)(9,14,15,11)(10,16,17,21)(12,19,22,20)(13,23,24,18)(25,30)(26,35)(27,37)(28,34)(29,32)(31,40)(33,39)(36,38)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=hi,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8806099);