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 | |
4658 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)(11,16)(12,21)(13,23)(14,17)(15,18)(19,20)(22,24)(25,26)(27,35)(28,39)(29,37)(30,36)(31,34)(32,38)(33,40);(3,7)(6,8)(11,18)(15,16)(19,22)(20,24)(25,26,31,34)(27,32,38,35)(28,39,36,30)(29,33,40,37);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,19)(13,20)(14,18)(16,17)(21,24)(22,23)(25,27,31,38)(26,32,34,35)(28,29,36,40)(30,37,39,33);(1,3)(2,6)(4,7)(5,8)(9,11)(10,16)(12,22)(13,20)(14,18)(15,17)(19,23)(21,24)(25,28)(26,30)(27,29)(31,36)(32,37)(33,35)(34,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,12)(10,13)(11,19)(14,23)(15,20)(16,24)(17,21)(18,22)(25,29)(26,33)(27,36)(28,38)(30,32)(31,40)(34,37)(35,39);(9,13)(10,12)(11,20)(14,21)(15,19)(16,22)(17,23)(18,24)(25,30)(26,28)(27,37)(29,32)(31,39)(33,38)(34,36)(35,40)Polycyclic presentation:
2^9|b^2=g,c^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8042532);