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 | |
4568 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(5,8)(9,10,14,13)(11,17,18,15)(12,16,20,21)(19,23,24,22)(25,26)(27,32)(28,31)(29,34)(30,33)(35,39)(36,37)(38,40);(1,2,4,6)(3,5,7,8)(10,13)(15,17)(16,21)(22,23)(27,34)(29,32)(35,36)(37,39);(1,3)(2,5)(4,7)(6,8)(11,18)(12,20)(15,17)(16,21)(25,27,30,34)(26,29,33,32)(28,36,38,35)(31,39,40,37);(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,24)(21,23)(25,28,30,38)(26,31,33,40)(27,35,34,36)(29,37,32,39);(1,2)(3,5)(4,6)(7,8)(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,27)(26,32)(28,35)(29,33)(30,34)(31,39)(36,38)(37,40);(1,3)(2,5)(4,7)(6,8)(9,13,14,10)(11,17,18,15)(12,21,20,16)(19,23,24,22)(25,29,30,32)(26,27,33,34)(28,37,38,39)(31,35,40,36)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=i,d^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8726186);