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 | |
4645 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,16)(12,13)(14,15)(17,22)(18,21)(19,23)(20,24)(26,31)(28,32)(33,39)(36,37);(1,2)(3,5)(4,6)(7,8)(9,10)(11,18)(12,13)(14,15)(16,21)(17,22)(19,20)(23,24)(25,26,30,31)(27,32,35,28)(29,33,38,39)(34,37,40,36);(2,3)(6,7)(10,15)(12,22)(16,18)(19,24)(25,27,30,35)(26,32,31,28)(29,34,38,40)(33,37,39,36);(4,8)(6,7)(9,11)(10,16)(12,19)(13,23)(14,21)(15,18)(17,20)(22,24)(25,28)(26,27)(29,37)(30,32)(31,35)(33,40)(34,39)(36,38);(1,3,5,2)(4,7,8,6)(9,12)(10,17)(11,19)(13,15)(14,22)(16,20)(18,23)(21,24)(25,27)(26,28)(29,34)(30,35)(31,32)(33,36)(37,39)(38,40);(1,4)(2,6)(3,7)(5,8)(9,13,14,17)(10,12,15,22)(11,20,21,23)(16,24,18,19)(25,29)(26,33)(27,34)(28,36)(30,38)(31,39)(32,37)(35,40)Polycyclic presentation:
2^9|b^2=g,c^2=g,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8521371);