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 | #47662 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4390 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)(3,5)(4,7)(6,8)(25,26,30,33)(27,34,29,31)(28,32,36,40)(35,39,37,38);(1,2,6,8)(3,5,7,4)(9,10)(11,15)(12,16)(13,14)(17,23)(18,21)(19,24)(20,22)(27,29)(31,34)(35,37)(38,39);(1,3,6,7)(2,4,8,5)(10,14)(15,21)(16,22)(23,24)(25,27,30,29)(26,31,33,34)(28,35,36,37)(32,38,40,39);(1,4,6,5)(2,3,8,7)(9,11,13,18)(10,15,14,21)(12,19,20,17)(16,24,22,23)(25,28)(26,32)(27,35)(29,37)(30,36)(31,38)(33,40)(34,39);(1,2)(3,4)(5,7)(6,8)(9,12,13,20)(10,16,14,22)(11,17,18,19)(15,23,21,24)(25,29,30,27)(26,31,33,34)(28,35,36,37)(32,39,40,38);(1,5,6,4)(2,7,8,3)(9,10)(11,15)(12,16)(13,14)(17,23)(18,21)(19,24)(20,22)(25,26)(27,34)(28,32)(29,31)(30,33)(35,39)(36,40)(37,38)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=h,d^2=ghi,e^2=gi,f^2=g,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, 8808419);