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 | #36410 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4516 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10,15,12)(11,16,19,20)(13,23,22,17)(14,18,21,24)(25,26)(27,35)(28,29)(30,33)(31,34)(32,36)(37,40)(38,39);(1,2)(3,6)(4,7)(5,8)(9,11)(10,16)(12,20)(13,21)(14,22)(15,19)(17,18)(23,24)(25,27,31,36)(26,32,34,35)(28,33,38,40)(29,37,39,30);(1,3,5,4)(2,6,8,7)(10,12)(11,19)(14,21)(17,23)(27,36)(30,37)(32,35)(33,40);(1,4,5,3)(2,7,8,6)(9,12,15,10)(11,20,19,16)(13,17,22,23)(14,24,21,18)(25,28,31,38)(26,29,34,39)(27,33,36,40)(30,32,37,35);(9,13,15,22)(10,17,12,23)(11,21,19,14)(16,18,20,24)(25,29)(26,28)(27,37)(30,36)(31,39)(32,33)(34,38)(35,40);(9,14,15,21)(10,18,12,24)(11,22,19,13)(16,17,20,23)(25,30,31,37)(26,33,34,40)(27,29,36,39)(28,32,38,35)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,c^2=ghi,d^2=gh,e^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7690418);