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 | |
4488 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,8,4,7)(9,10)(11,16)(12,13)(14,22)(15,17)(18,24)(19,21)(20,23)(25,26,31,29)(27,30,35,32)(28,36,37,33)(34,39,40,38);(2,6)(7,8)(9,11,15,20)(10,16,17,23)(12,14,18,21)(13,22,24,19)(25,27,31,35)(26,32,29,30)(28,34,37,40)(33,39,36,38);(2,6)(7,8)(10,17)(11,20)(12,18)(19,22)(28,37)(33,36)(34,40)(38,39);(1,2)(3,7)(4,8)(5,6)(9,12,15,18)(10,13,17,24)(11,21,20,14)(16,19,23,22)(25,28)(26,33)(27,34)(29,36)(30,38)(31,37)(32,39)(35,40);(1,3,5,4)(2,7,6,8)(9,13)(10,18)(11,22)(12,17)(14,23)(15,24)(16,21)(19,20)(25,29,31,26)(27,30,35,32)(28,36,37,33)(34,38,40,39);(1,4,5,3)(2,8,6,7)(9,14,15,21)(10,19,17,22)(11,18,20,12)(13,23,24,16)(25,30)(26,27)(28,38)(29,35)(31,32)(33,34)(36,40)(37,39)Polycyclic presentation:
2^9|a^2=g,b^2=gh,d^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8014523);