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 | |
4730 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)(6,8)(9,10)(11,14)(12,17)(13,18)(15,20)(16,21)(19,22)(23,24)(25,26)(27,32)(28,34)(29,38)(30,37)(31,33)(35,39)(36,40);(3,7)(6,8)(10,11)(15,17)(16,18)(22,23)(25,27)(26,32)(28,31)(29,37)(30,38)(33,34)(35,40)(36,39);(1,2)(3,6)(4,5)(7,8)(10,11)(15,17)(16,18)(22,23)(26,33)(27,28)(35,38)(36,37);(1,3)(2,6)(4,7)(5,8)(9,11,14,10)(12,15,20,17)(13,18,21,16)(19,22,24,23)(25,28,31,27)(26,34,33,32)(29,37,39,36)(30,35,40,38);(1,2)(3,6)(4,5)(7,8)(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,29)(26,35)(27,37)(28,36)(30,34)(31,39)(32,40)(33,38);(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(25,30)(26,36)(27,38)(28,35)(29,34)(31,40)(32,39)(33,37)Polycyclic presentation:
2^9|d^2=gi,e^2=g,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^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8066529);