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 | #47694 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4342 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)(3,8)(4,7)(5,6)(9,10,14,15)(11,19,20,16)(12,17,22,21)(13,23,24,18);(3,4)(7,8)(10,15)(12,22)(16,19)(18,23)(25,26,30,28)(27,31,33,35)(29,32,37,36)(34,38,39,40);(1,3)(2,7)(4,6)(5,8)(10,15)(11,20)(12,22)(13,24)(26,28)(27,33)(29,37)(38,40);(3,4)(7,8)(9,11,14,20)(10,16,15,19)(12,23,22,18)(13,21,24,17)(25,27)(26,31)(28,35)(29,34)(30,33)(32,38)(36,40)(37,39);(1,4,6,3)(2,7,5,8)(9,12)(10,17)(11,18)(13,19)(14,22)(15,21)(16,24)(20,23)(25,28,30,26)(27,31,33,35)(29,36,37,32)(34,38,39,40);(1,5,6,2)(3,7,4,8)(9,13,14,24)(10,18,15,23)(11,21,20,17)(12,19,22,16)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,d^2=g,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8911131);