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 | |
4507 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(5,8)(10,15)(11,17)(16,23)(18,20)(25,26,30,34)(27,28,32,31)(29,37,39,33)(35,40,38,36);(1,2,4,6)(3,5,7,8)(9,10)(11,13)(12,16)(14,15)(17,22)(18,21)(19,23)(20,24)(25,27)(26,28)(29,35)(30,32)(31,34)(33,36)(37,40)(38,39);(1,3)(2,5)(4,7)(6,8)(11,17)(12,19)(13,22)(16,23)(25,27,30,32)(26,31,34,28)(29,38,39,35)(33,36,37,40);(9,11)(10,13)(12,20)(14,17)(15,22)(16,24)(18,19)(21,23)(25,27)(26,28)(29,35)(30,32)(31,34)(33,36)(37,40)(38,39);(1,2)(3,5)(4,6)(7,8)(9,12)(10,16)(11,18)(13,21)(14,19)(15,23)(17,20)(22,24)(25,28)(26,32)(27,34)(29,36)(30,31)(33,38)(35,37)(39,40);(1,3)(2,5)(4,7)(6,8)(9,13)(10,11)(12,21)(14,22)(15,17)(16,18)(19,24)(20,23)(25,29,30,39)(26,33,34,37)(27,35,32,38)(28,36,31,40)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=i,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8936256);