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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4600 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(3,8)(9,10)(11,14)(12,17)(13,18)(15,19)(16,21)(20,22)(23,24)(25,26)(27,32)(28,30)(29,36)(31,34)(33,37)(35,38)(39,40);(1,2,6,5)(3,7,8,4)(10,11)(15,17)(16,18)(22,23)(27,34)(31,32)(35,40)(38,39);(3,8)(4,7)(10,11)(12,19)(16,18)(20,24)(25,27,30,34)(26,31,28,32)(29,35,37,40)(33,38,36,39);(1,3,6,8)(2,7,5,4)(9,11,14,10)(12,17,19,15)(13,18,21,16)(20,23,24,22)(25,27)(26,32)(28,31)(29,35)(30,34)(33,39)(36,38)(37,40);(1,4)(2,3)(5,8)(6,7)(9,12,14,19)(10,15,11,17)(13,20,21,24)(16,22,18,23)(25,28,30,26)(27,31,34,32)(29,36,37,33)(35,38,40,39);(1,5,6,2)(3,4,8,7)(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40)Polycyclic presentation:
2^9|b^2=ghi,c^2=i,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8034608);